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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)(-10,68)
158 \includegraphics[width=2.6cm]{wide_band_gap.eps}
160 \begin{picture}(0,0)(-295,-165)
161 \includegraphics[width=3cm]{sic_led.eps}
163 \begin{picture}(0,0)(-215,-165)
164 \includegraphics[width=2.5cm]{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}
184 \item Polytyps and fabrication of silicon carbide
185 \item Supposed precipitation mechanism of SiC in Si
186 \item Utilized simulation techniques
188 \item Molecular dynamics (MD) simulations
189 \item Density functional theory (DFT) calculations
191 \item C and Si self-interstitial point defects in silicon
192 \item Silicon carbide precipitation simulations
193 \item Investigation of a silicon carbide precipitate in silicon
194 \item Summary / Conclusion / Outlook
211 \begin{tabular}{l c c c c c c}
213 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
215 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
216 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
217 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
218 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
219 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
220 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
221 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
229 \begin{picture}(0,0)(-160,-155)
230 \includegraphics[width=7cm]{polytypes.eps}
232 \begin{picture}(0,0)(-10,-185)
233 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
235 \begin{picture}(0,0)(-10,-175)
236 {\tiny cubic (twist)}
238 \begin{picture}(0,0)(-60,-175)
239 {\tiny hexagonal (no twist)}
241 \begin{pspicture}(0,0)(0,0)
242 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
244 \begin{pspicture}(0,0)(0,0)
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247 \begin{pspicture}(0,0)(0,0)
248 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
256 Fabrication of silicon carbide
263 SiC - \emph{Born from the stars, perfected on earth.}
267 Conventional thin film SiC growth:
269 \item \underline{Sublimation growth using the modified Lely method}
271 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
272 \item Surrounded by polycrystalline SiC in a graphite crucible\\
273 at $T=2100-2400 \, ^{\circ} \text{C}$
274 \item Deposition of supersaturated vapor on cooler seed crystal
276 \item \underline{Homoepitaxial growth using CVD}
278 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
279 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
280 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
281 \item High quality but limited in size of substrates
283 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
285 \item Two steps: carbonization and growth
286 \item $T=650-1050 \, ^{\circ} \text{C}$
287 \item Quality and size not yet sufficient
291 \begin{picture}(0,0)(-280,-65)
292 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
294 \begin{picture}(0,0)(-280,-55)
295 \begin{minipage}{5cm}
297 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
302 \begin{picture}(0,0)(-265,-150)
303 \includegraphics[width=2.4cm]{m_lely.eps}
305 \begin{picture}(0,0)(-333,-175)
306 \begin{minipage}{5cm}
312 5. Insulation\\[-7pt]
323 Fabrication of silicon carbide
328 Alternative approach:
329 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
331 \item \underline{Implantation step 1}\\
332 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
333 $\Rightarrow$ box-like distribution of equally sized
334 and epitactically oriented SiC precipitates
336 \item \underline{Implantation step 2}\\
337 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
338 $\Rightarrow$ destruction of SiC nanocrystals
339 in growing amorphous interface layers
340 \item \underline{Annealing}\\
341 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
342 $\Rightarrow$ homogeneous, stoichiometric SiC layer
343 with sharp interfaces
346 \begin{minipage}{6.3cm}
347 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
349 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
353 \begin{minipage}{6.3cm}
356 Precipitation mechanism not yet fully understood!
358 \renewcommand\labelitemi{$\Rightarrow$}
360 \underline{Understanding the SiC precipitation}
362 \item significant technological progress in SiC thin film formation
363 \item perspectives for processes relying upon prevention of SiC precipitation
374 Supposed precipitation mechanism of SiC in Si
381 \begin{minipage}{3.8cm}
382 Si \& SiC lattice structure\\[0.2cm]
383 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
387 \begin{minipage}{3.8cm}
389 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
393 \begin{minipage}{3.8cm}
395 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
399 \begin{minipage}{4cm}
401 C-Si dimers (dumbbells)\\[-0.1cm]
402 on Si interstitial sites
406 \begin{minipage}{4.2cm}
408 Agglomeration of C-Si dumbbells\\[-0.1cm]
409 $\Rightarrow$ dark contrasts
413 \begin{minipage}{4cm}
415 Precipitation of 3C-SiC in Si\\[-0.1cm]
416 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
417 \& release of Si self-interstitials
421 \begin{minipage}{3.8cm}
423 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
427 \begin{minipage}{3.8cm}
429 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
433 \begin{minipage}{3.8cm}
435 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
439 \begin{pspicture}(0,0)(0,0)
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451 Molecular dynamics (MD) simulations
460 \item Microscopic description of N particle system
461 \item Analytical interaction potential
462 \item Numerical integration using Newtons equation of motion\\
463 as a propagation rule in 6N-dimensional phase space
464 \item Observables obtained by time and/or ensemble averages
466 {\bf Details of the simulation:}
468 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
469 \item Ensemble: NpT (isothermal-isobaric)
471 \item Berendsen thermostat:
472 $\tau_{\text{T}}=100\text{ fs}$
473 \item Berendsen barostat:\\
474 $\tau_{\text{P}}=100\text{ fs}$,
475 $\beta^{-1}=100\text{ GPa}$
477 \item Erhart/Albe potential: Tersoff-like bond order potential
480 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
481 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
485 \begin{picture}(0,0)(-230,-30)
486 \includegraphics[width=5cm]{tersoff_angle.eps}
494 Density functional theory (DFT) calculations
499 Basic ingredients necessary for DFT
502 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
504 \item ... uniquely determines the ground state potential
506 \item ... minimizes the systems total energy
508 \item \underline{Born-Oppenheimer}
509 - $N$ moving electrons in an external potential of static nuclei
511 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
512 +\sum_i^N V_{\text{ext}}(r_i)
513 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
515 \item \underline{Effective potential}
516 - averaged electrostatic potential \& exchange and correlation
518 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
521 \item \underline{Kohn-Sham system}
522 - Schr\"odinger equation of N non-interacting particles
524 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
529 n(r)=\sum_i^N|\Phi_i(r)|^2
531 \item \underline{Self-consistent solution}\\
532 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
533 which in turn depends on $n(r)$
534 \item \underline{Variational principle}
535 - minimize total energy with respect to $n(r)$
543 Density functional theory (DFT) calculations
550 Details of applied DFT calculations in this work
553 \item \underline{Exchange correlation functional}
554 - approximations for the inhomogeneous electron gas
556 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
557 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
559 \item \underline{Plane wave basis set}
560 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
563 \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}}
565 \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
566 \item \underline{Pseudo potential}
567 - consider only the valence electrons
568 \item \underline{Code} - VASP 4.6
573 MD and structural optimization
576 \item MD integration: Gear predictor corrector algorithm
577 \item Pressure control: Parrinello-Rahman pressure control
578 \item Structural optimization: Conjugate gradient method
586 C and Si self-interstitial point defects in silicon
593 \begin{minipage}{8cm}
595 \begin{pspicture}(0,0)(7,5)
596 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
599 \item Creation of c-Si simulation volume
600 \item Periodic boundary conditions
601 \item $T=0\text{ K}$, $p=0\text{ bar}$
604 \rput(3.5,2.1){\rnode{insert}{\psframebox{
607 Insertion of interstitial C/Si atoms
610 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
613 Relaxation / structural energy minimization
616 \ncline[]{->}{init}{insert}
617 \ncline[]{->}{insert}{cool}
620 \begin{minipage}{5cm}
621 \includegraphics[width=5cm]{unit_cell_e.eps}\\
624 \begin{minipage}{9cm}
625 \begin{tabular}{l c c}
627 & size [unit cells] & \# atoms\\
629 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
630 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
634 \begin{minipage}{4cm}
635 {\color{red}$\bullet$} Tetrahedral\\
636 {\color{green}$\bullet$} Hexagonal\\
637 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
638 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
639 {\color{cyan}$\bullet$} Bond-centered\\
640 {\color{black}$\bullet$} Vacancy / Substitutional
649 \begin{minipage}{9.5cm}
652 Si self-interstitial point defects in silicon\\
655 \begin{tabular}{l c c c c c}
657 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
659 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
660 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
662 \end{tabular}\\[0.2cm]
664 \begin{minipage}{4.7cm}
665 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
667 \begin{minipage}{4.7cm}
669 {\tiny nearly T $\rightarrow$ T}\\
671 \includegraphics[width=4.7cm]{nhex_tet.ps}
674 \underline{Hexagonal} \hspace{2pt}
675 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
677 \begin{minipage}{2.7cm}
678 $E_{\text{f}}^*=4.48\text{ eV}$\\
679 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
681 \begin{minipage}{0.4cm}
686 \begin{minipage}{2.7cm}
687 $E_{\text{f}}=3.96\text{ eV}$\\
688 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
691 \begin{minipage}{2.9cm}
693 \underline{Vacancy}\\
694 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
699 \begin{minipage}{3.5cm}
702 \underline{\hkl<1 1 0> dumbbell}\\
703 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
704 \underline{Tetrahedral}\\
705 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
706 \underline{\hkl<1 0 0> dumbbell}\\
707 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
719 C interstitial point defects in silicon\\[-0.1cm]
722 \begin{tabular}{l c c c c c c}
724 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
726 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
727 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
729 \end{tabular}\\[0.1cm]
732 \begin{minipage}{2.7cm}
733 \underline{Hexagonal} \hspace{2pt}
734 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
735 $E_{\text{f}}^*=9.05\text{ eV}$\\
736 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
738 \begin{minipage}{0.4cm}
743 \begin{minipage}{2.7cm}
744 \underline{\hkl<1 0 0>}\\
745 $E_{\text{f}}=3.88\text{ eV}$\\
746 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
749 \begin{minipage}{2cm}
752 \begin{minipage}{3cm}
754 \underline{Tetrahedral}\\
755 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
760 \begin{minipage}{2.7cm}
761 \underline{Bond-centered}\\
762 $E_{\text{f}}^*=5.59\text{ eV}$\\
763 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
765 \begin{minipage}{0.4cm}
770 \begin{minipage}{2.7cm}
771 \underline{\hkl<1 1 0> dumbbell}\\
772 $E_{\text{f}}=5.18\text{ eV}$\\
773 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
776 \begin{minipage}{2cm}
779 \begin{minipage}{3cm}
781 \underline{Substitutional}\\
782 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
793 C \hkl<1 0 0> dumbbell interstitial configuration\\
797 \begin{tabular}{l c c c c c c c c}
799 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
801 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
802 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
804 \end{tabular}\\[0.2cm]
805 \begin{tabular}{l c c c c }
807 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
809 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
810 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
812 \end{tabular}\\[0.2cm]
813 \begin{tabular}{l c c c}
815 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
817 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
818 VASP & 0.109 & -0.065 & 0.174 \\
820 \end{tabular}\\[0.6cm]
823 \begin{minipage}{3.0cm}
825 \underline{Erhart/Albe}
826 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
829 \begin{minipage}{3.0cm}
832 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
836 \begin{picture}(0,0)(-185,10)
837 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
839 \begin{picture}(0,0)(-280,-150)
840 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
843 \begin{pspicture}(0,0)(0,0)
844 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
845 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
846 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
847 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
856 \begin{minipage}{8.5cm}
859 Bond-centered interstitial configuration\\[-0.1cm]
862 \begin{minipage}{3.0cm}
863 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
865 \begin{minipage}{5.2cm}
867 \item Linear Si-C-Si bond
868 \item Si: one C \& 3 Si neighbours
869 \item Spin polarized calculations
870 \item No saddle point!\\
877 \begin{minipage}[t]{6.5cm}
878 \begin{minipage}[t]{1.2cm}
880 {\tiny sp$^3$}\\[0.8cm]
881 \underline{${\color{black}\uparrow}$}
882 \underline{${\color{black}\uparrow}$}
883 \underline{${\color{black}\uparrow}$}
884 \underline{${\color{red}\uparrow}$}\\
887 \begin{minipage}[t]{1.4cm}
889 {\color{red}M}{\color{blue}O}\\[0.8cm]
890 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
891 $\sigma_{\text{ab}}$\\[0.5cm]
892 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
896 \begin{minipage}[t]{1.0cm}
900 \underline{${\color{white}\uparrow\uparrow}$}
901 \underline{${\color{white}\uparrow\uparrow}$}\\
903 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
904 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
908 \begin{minipage}[t]{1.4cm}
910 {\color{blue}M}{\color{green}O}\\[0.8cm]
911 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
912 $\sigma_{\text{ab}}$\\[0.5cm]
913 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
917 \begin{minipage}[t]{1.2cm}
920 {\tiny sp$^3$}\\[0.8cm]
921 \underline{${\color{green}\uparrow}$}
922 \underline{${\color{black}\uparrow}$}
923 \underline{${\color{black}\uparrow}$}
924 \underline{${\color{black}\uparrow}$}\\
932 \begin{minipage}{4.5cm}
933 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
935 \begin{minipage}{3.5cm}
936 {\color{gray}$\bullet$} Spin up\\
937 {\color{green}$\bullet$} Spin down\\
938 {\color{blue}$\bullet$} Resulting spin up\\
939 {\color{yellow}$\bullet$} Si atoms\\
940 {\color{red}$\bullet$} C atom
945 \begin{minipage}{4.2cm}
947 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
948 {\color{green}$\Box$} {\tiny unoccupied}\\
949 {\color{red}$\bullet$} {\tiny occupied}
958 Migration of the C \hkl<1 0 0> dumbbell interstitial
963 {\small Investigated pathways}
965 \begin{minipage}{8.5cm}
966 \begin{minipage}{8.3cm}
967 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
968 \begin{minipage}{2.4cm}
969 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
971 \begin{minipage}{0.4cm}
974 \begin{minipage}{2.4cm}
975 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
977 \begin{minipage}{0.4cm}
980 \begin{minipage}{2.4cm}
981 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
984 \begin{minipage}{8.3cm}
985 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
986 \begin{minipage}{2.4cm}
987 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
989 \begin{minipage}{0.4cm}
992 \begin{minipage}{2.4cm}
993 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
995 \begin{minipage}{0.4cm}
998 \begin{minipage}{2.4cm}
999 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1002 \begin{minipage}{8.3cm}
1003 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1004 \begin{minipage}{2.4cm}
1005 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1007 \begin{minipage}{0.4cm}
1010 \begin{minipage}{2.4cm}
1011 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1013 \begin{minipage}{0.4cm}
1016 \begin{minipage}{2.4cm}
1017 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1022 \begin{minipage}{4.2cm}
1023 {\small Constrained relaxation\\
1024 technique (CRT) method}\\
1025 \includegraphics[width=4cm]{crt_orig.eps}
1027 \item Constrain diffusing atom
1028 \item Static constraints
1031 {\small Modifications}\\
1032 \includegraphics[width=4cm]{crt_mod.eps}
1034 \item Constrain all atoms
1035 \item Update individual\\
1046 Migration of the C \hkl<1 0 0> dumbbell interstitial
1052 \begin{minipage}{5.9cm}
1054 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
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1076 \begin{minipage}{0.3cm}
1080 \begin{minipage}{5.9cm}
1082 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
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1097 \begin{picture}(0,0)(90,0)
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1108 \begin{minipage}{5.9cm}
1110 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
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1123 \includegraphics[width=1cm]{100_arrow.eps}
1125 \begin{picture}(0,0)(90,0)
1126 \includegraphics[height=0.9cm]{001_arrow.eps}
1132 \begin{minipage}{0.3cm}
1135 \begin{minipage}{6.5cm}
1138 \item Energetically most favorable path
1141 \item Activation energy: $\approx$ 0.9 eV
1142 \item Experimental values: 0.73 ... 0.87 eV
1144 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1145 \item Reorientation (path 3)
1147 \item More likely composed of two consecutive steps of type 2
1148 \item Experimental values: 0.77 ... 0.88 eV
1150 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1159 Migration of the C \hkl<1 0 0> dumbbell interstitial
1164 \begin{minipage}{6.5cm}
1167 \begin{minipage}{5.9cm}
1169 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1172 \begin{pspicture}(0,0)(0,0)
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1178 \begin{picture}(0,0)(5,-50)
1179 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
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1182 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
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1185 \includegraphics[width=1cm]{110_arrow.eps}
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1197 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1199 \begin{picture}(0,0)(-5,-15)
1200 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
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1203 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1205 \begin{picture}(0,0)(12.5,-5)
1206 \includegraphics[width=1cm]{100_arrow.eps}
1208 \begin{picture}(0,0)(90,-15)
1209 \includegraphics[height=0.9cm]{010_arrow.eps}
1215 \begin{minipage}{5.9cm}
1218 \item Lowest activation energy: $\approx$ 2.2 eV
1219 \item 2.4 times higher than VASP
1220 \item Different pathway
1221 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1226 \begin{minipage}{6.5cm}
1229 \begin{minipage}{5.9cm}
1231 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1234 \begin{pspicture}(0,0)(0,0)
1235 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1237 \begin{picture}(0,0)(60,-5)
1238 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1240 \begin{picture}(0,0)(0,-5)
1241 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
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1244 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
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1247 \includegraphics[width=1cm]{100_arrow.eps}
1249 \begin{picture}(0,0)(90,0)
1250 \includegraphics[height=0.9cm]{001_arrow.eps}
1258 \begin{minipage}{5.9cm}
1259 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1270 Migrations involving the C \hkl<1 1 0> dumbbell interstitial
1279 \begin{minipage}{6.0cm}
1280 \includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
1282 \begin{minipage}{7cm}
1283 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1284 \hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
1285 \hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
1286 BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
1287 Composed of three single transitions\\[0.3cm]
1288 Activation energy of second transition slightly\\
1289 higher than direct transition (path 2)\\[0.3cm]
1290 $\Rightarrow$ very unlikely to happen
1291 \end{minipage}\\[0.2cm]
1295 \begin{minipage}{6.0cm}
1296 \includegraphics[width=6cm]{110_mig.ps}
1298 \begin{minipage}{7cm}
1299 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1300 \hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
1301 \hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
1302 \hkl<0 0 -1>\\[0.3cm]
1303 Composed of two single transitions\\[0.3cm]
1304 Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
1305 $\Rightarrow$ more readily constituting a probable transition
1313 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1323 E_{\text{f}}^{\text{defect combination}}-
1324 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1325 E_{\text{f}}^{\text{2nd defect}}
1331 \begin{tabular}{l c c c c c c}
1333 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1335 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1336 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1337 \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}\\
1338 \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}\\
1339 \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}\\
1340 \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}\\
1342 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1343 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1352 \begin{minipage}[t]{3.8cm}
1353 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1354 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1356 \begin{minipage}[t]{3.5cm}
1357 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1358 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1360 \begin{minipage}[t]{5.5cm}
1362 \item Restricted to VASP simulations
1363 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1364 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1365 \item Stress compensation / increase
1366 \item Most favorable: C clustering
1367 \item Unfavored: antiparallel orientations
1368 \item Indication of energetically favored\\
1373 \begin{picture}(0,0)(-295,-130)
1374 \includegraphics[width=3.5cm]{comb_pos.eps}
1382 Combinations of C-Si \hkl<1 0 0>-type interstitials
1389 Energetically most favorable combinations along \hkl<1 1 0>
1394 \begin{tabular}{l c c c c c c}
1396 & 1 & 2 & 3 & 4 & 5 & 6\\
1398 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1399 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1400 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>\\
1407 \begin{minipage}{7.0cm}
1408 \includegraphics[width=7cm]{db_along_110_cc.ps}
1410 \begin{minipage}{6.0cm}
1413 Interaction proportional to reciprocal cube of C-C distance
1415 Saturation in the immediate vicinity
1426 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1432 \begin{minipage}{3.2cm}
1433 \includegraphics[width=3cm]{sub_110_combo.eps}
1435 \begin{minipage}{7.8cm}
1436 \begin{tabular}{l c c c c c c}
1438 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1439 \hkl<1 0 1> & \hkl<-1 0 1> \\
1441 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1442 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1443 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1444 4 & \RM{4} & B & D & E & E & D \\
1445 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1452 \begin{tabular}{l c c c c c c c c c c}
1454 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1456 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1457 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1458 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1463 \begin{minipage}{6.0cm}
1464 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1466 \begin{minipage}{7cm}
1469 \item IBS: C may displace Si\\
1470 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1472 \hkl<1 1 0>-type $\rightarrow$ favored combination
1473 \renewcommand\labelitemi{$\Rightarrow$}
1474 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1475 ($E_{\text{f}}=3.88\text{ eV}$)
1476 \item Interaction drops quickly to zero\\
1477 (low interaction capture radius)
1486 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1493 \begin{minipage}[t]{3cm}
1494 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1495 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1497 \begin{minipage}[t]{7cm}
1500 Low activation energies\\
1501 High activation energies for reverse processes\\
1503 {\color{blue}C$_{\text{sub}}$ very stable}\\
1507 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1509 {\color{blue}Formation of SiC by successive substitution by C}
1513 \begin{minipage}[t]{3cm}
1514 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1515 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1520 \begin{minipage}{5.9cm}
1521 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1523 \begin{picture}(0,0)(70,0)
1524 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1526 \begin{picture}(0,0)(30,0)
1527 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1529 \begin{picture}(0,0)(-10,0)
1530 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1532 \begin{picture}(0,0)(-48,0)
1533 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1535 \begin{picture}(0,0)(12.5,5)
1536 \includegraphics[width=1cm]{100_arrow.eps}
1538 \begin{picture}(0,0)(97,-10)
1539 \includegraphics[height=0.9cm]{001_arrow.eps}
1545 \begin{minipage}{0.3cm}
1549 \begin{minipage}{5.9cm}
1550 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1552 \begin{picture}(0,0)(60,0)
1553 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1555 \begin{picture}(0,0)(25,0)
1556 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1558 \begin{picture}(0,0)(-20,0)
1559 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1561 \begin{picture}(0,0)(-55,0)
1562 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1564 \begin{picture}(0,0)(12.5,5)
1565 \includegraphics[width=1cm]{100_arrow.eps}
1567 \begin{picture}(0,0)(95,0)
1568 \includegraphics[height=0.9cm]{001_arrow.eps}
1580 Conclusion of defect / migration / combined defect simulations
1589 \item Accurately described by quantum-mechanical simulations
1590 \item Less correct description by classical potential simulations
1594 \item Consistent with solubility data of C in Si
1595 \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration
1596 \item Consistent with reorientation and diffusion experiments
1597 \item C migration pathway in Si identified
1602 Concerning the precipitation mechanism
1604 \item Agglomeration of C-Si dumbbells energetically favorable
1605 \item C-Si indeed favored compared to
1606 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1607 \item Possible low interaction capture radius of
1608 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1609 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1610 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1615 {\color{blue}Some results point to a different precipitation mechanism!}
1623 Silicon carbide precipitation simulations
1629 \begin{pspicture}(0,0)(12,6.5)
1631 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1634 \item Create c-Si volume
1635 \item Periodc boundary conditions
1636 \item Set requested $T$ and $p=0\text{ bar}$
1637 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1640 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1642 Insertion of C atoms at constant T
1644 \item total simulation volume {\pnode{in1}}
1645 \item volume of minimal SiC precipitate {\pnode{in2}}
1646 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1650 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1652 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1654 \ncline[]{->}{init}{insert}
1655 \ncline[]{->}{insert}{cool}
1656 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1657 \rput(7.8,6){\footnotesize $V_1$}
1658 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1659 \rput(9.2,4.85){\tiny $V_2$}
1660 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1661 \rput(9.55,4.45){\footnotesize $V_3$}
1662 \rput(7.9,3.2){\pnode{ins1}}
1663 \rput(9.22,2.8){\pnode{ins2}}
1664 \rput(11.0,2.4){\pnode{ins3}}
1665 \ncline[]{->}{in1}{ins1}
1666 \ncline[]{->}{in2}{ins2}
1667 \ncline[]{->}{in3}{ins3}
1672 \item Restricted to classical potential simulations
1673 \item $V_2$ and $V_3$ considered due to low diffusion
1674 \item Amount of C atoms: 6000
1675 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1676 \item Simulation volume: $31\times 31\times 31$ unit cells
1685 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1690 \begin{minipage}{6.5cm}
1691 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1693 \begin{minipage}{6.5cm}
1694 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1697 \begin{minipage}{6.5cm}
1698 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1700 \begin{minipage}{6.5cm}
1702 \underline{Low C concentration ($V_1$)}\\
1703 \hkl<1 0 0> C-Si dumbbell dominated structure
1705 \item Si-C bumbs around 0.19 nm
1706 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1707 concatenated dumbbells of various orientation
1708 \item Si-Si NN distance stretched to 0.3 nm
1710 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1711 \underline{High C concentration ($V_2$, $V_3$)}\\
1712 High amount of strongly bound C-C bonds\\
1713 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1714 Only short range order observable\\
1715 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1723 Limitations of molecular dynamics and short range potentials
1730 \underline{Time scale problem of MD}\\[0.2cm]
1731 Minimize integration error\\
1732 $\Rightarrow$ discretization considerably smaller than
1733 reciprocal of fastest vibrational mode\\[0.1cm]
1734 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1735 $\Rightarrow$ suitable choice of time step:
1736 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1737 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1738 Several local minima in energy surface separated by large energy barriers\\
1739 $\Rightarrow$ transition event corresponds to a multiple
1740 of vibrational periods\\
1741 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1742 infrequent transition events\\[0.1cm]
1743 {\color{blue}Accelerated methods:}
1744 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1748 \underline{Limitations related to the short range potential}\\[0.2cm]
1749 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1750 and 2$^{\text{nd}}$ next neighbours\\
1751 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1757 Potential enhanced problem of slow phase space propagation
1762 \underline{Approach to the (twofold) problem}\\[0.2cm]
1763 Increased temperature simulations without TAD corrections\\
1764 (accelerated methods or higher time scales exclusively not sufficient)
1766 \begin{picture}(0,0)(-260,-30)
1768 \begin{minipage}{4.2cm}
1775 \item 3C-SiC also observed for higher T
1776 \item higher T inside sample
1777 \item structural evolution vs.\\
1778 equilibrium properties
1784 \begin{picture}(0,0)(-305,-155)
1786 \begin{minipage}{2.5cm}
1790 thermodynmic sampling
1801 Increased temperature simulations at low C concentration
1806 \begin{minipage}{6.5cm}
1807 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1809 \begin{minipage}{6.5cm}
1810 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1813 \begin{minipage}{6.5cm}
1814 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1816 \begin{minipage}{6.5cm}
1818 \underline{Si-C bonds:}
1820 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1821 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1823 \underline{Si-Si bonds:}
1824 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1825 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1826 \underline{C-C bonds:}
1828 \item C-C next neighbour pairs reduced (mandatory)
1829 \item Peak at 0.3 nm slightly shifted
1831 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1832 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1834 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1836 \item Range [|-$\downarrow$]:
1837 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1838 with nearby Si$_{\text{I}}$}
1843 \begin{picture}(0,0)(-330,-74)
1846 \begin{minipage}{1.6cm}
1849 stretched SiC\\[-0.1cm]
1861 Increased temperature simulations at high C concentration
1866 \begin{minipage}{6.5cm}
1867 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1869 \begin{minipage}{6.5cm}
1870 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1874 Decreasing cut-off artifact\\
1875 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1876 $\Rightarrow$ hard to categorize
1882 \begin{minipage}[t]{6.0cm}
1883 0.186 nm: Si-C pairs $\uparrow$\\
1884 (as expected in 3C-SiC)\\[0.2cm]
1885 0.282 nm: Si-C-C\\[0.2cm]
1886 $\approx$0.35 nm: C-Si-Si
1889 \begin{minipage}{0.2cm}
1893 \begin{minipage}[t]{6.0cm}
1894 0.15 nm: C-C pairs $\uparrow$\\
1895 (as expected in graphite/diamond)\\[0.2cm]
1896 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1897 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1904 {\color{red}Amorphous} SiC-like phase remains\\
1905 Slightly sharper peaks
1906 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
1907 due to temperature\\[0.1cm]
1910 Continue with higher temperatures and longer time scales
1919 Valuation of a practicable temperature limit
1929 Recrystallization is a hard task!
1930 $\Rightarrow$ Avoid melting!
1939 \begin{minipage}{7.5cm}
1940 \includegraphics[width=7cm]{fe_and_t.ps}
1942 \begin{minipage}{5.5cm}
1943 \underline{Melting does not occur instantly after}\\
1944 \underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
1946 \item required transition enthalpy
1947 \item hysterisis behaviour
1949 \underline{Heating up c-Si by 1 K/ps}
1951 \item transition occurs at $\approx$ 3125 K
1952 \item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
1953 (literature: 50.2 kJ/mole)
1960 \begin{minipage}{4cm}
1961 Initially chosen temperatures:\\
1962 $1.0 - 1.2 \cdot T_{\text{m}}$
1965 \begin{minipage}{3cm}
1971 \begin{minipage}{5cm}
1972 Introduced C (defects)\\
1973 $\rightarrow$ reduction of transition point\\
1974 $\rightarrow$ melting even at $T_{\text{m}}$
1983 Maximum temperature used: $0.95\cdot T_{\text{m}}$
1993 Long time scale simulations at maximum temperature
2000 \underline{Differences}
2002 \item Cubic volume $\Rightarrow$ spherical volume
2003 \item Amount of C atoms: 6000 $\rightarrow$ 5500
2004 \item Temperature set to $0.95 \cdot T_{\text{m}}$
2005 \item Simulation volume: 21 unit cells of c-Si in each direction
2012 Simulations in progress! :)\\
2014 ... show evolution of radial distribution in ns timesteps ...
2024 Investigation of a silicon carbide precipitate in silicon
2033 \begin{minipage}{5.3cm}
2035 \frac{8}{a_{\text{Si}}^3}(
2036 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
2037 -\frac{4}{3}\pi x^3)+
2038 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
2045 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
2046 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
2049 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
2053 \begin{minipage}{0.3cm}
2056 \begin{minipage}{7.0cm}
2057 \underline{Construction}
2059 \item Simulation volume: 21$^3$ unit cells of c-Si
2060 \item Spherical topotactically aligned precipitate\\
2061 $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
2062 \item Create c-Si but skipped inside sphere of radius $x$
2063 \item Create 3C-SiC inside sphere of radius $x$ and lattice constant $y$
2064 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
2070 \begin{minipage}{6.2cm}
2071 \includegraphics[width=6cm,draft=false]{pc_0.ps}
2073 \begin{minipage}{6.8cm}
2076 \item Slight increase of c-Si lattice constant!
2077 \item C-C peaks (imply same distanced Si-Si peaks)
2079 \item New peak at 0.307 nm: NN in 3C-SiC
2080 \item Bumps ({\color{green}$\downarrow$}):
2081 4$^{\text{th}}$ and 6$^{\text{th}}$ NN
2083 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
2084 $\rightarrow$ compressed precipitate
2085 \item Interface tension:\\
2086 20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$
2095 Summary / Conclusion / Outlook