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96 Atomistic simulation study of the silicon carbide precipitation
102 \textsc{F. Zirkelbach}
115 % motivation / properties / applications of silicon carbide
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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]
317 \begin{picture}(0,0)(-230,-35)
319 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
322 \begin{picture}(0,0)(-230,-10)
324 \begin{minipage}{3cm}
325 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
336 Fabrication of silicon carbide
341 Alternative approach:
342 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
344 \item \underline{Implantation step 1}\\
345 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
346 $\Rightarrow$ box-like distribution of equally sized
347 and epitactically oriented SiC precipitates
349 \item \underline{Implantation step 2}\\
350 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
351 $\Rightarrow$ destruction of SiC nanocrystals
352 in growing amorphous interface layers
353 \item \underline{Annealing}\\
354 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
355 $\Rightarrow$ homogeneous, stoichiometric SiC layer
356 with sharp interfaces
359 \begin{minipage}{6.3cm}
360 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
362 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
366 \begin{minipage}{6.3cm}
369 Precipitation mechanism not yet fully understood!
371 \renewcommand\labelitemi{$\Rightarrow$}
373 \underline{Understanding the SiC precipitation}
375 \item significant technological progress in SiC thin film formation
376 \item perspectives for processes relying upon prevention of SiC precipitation
387 Supposed precipitation mechanism of SiC in Si
394 \begin{minipage}{3.8cm}
395 Si \& SiC lattice structure\\[0.2cm]
396 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
400 \begin{minipage}{3.8cm}
402 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
406 \begin{minipage}{3.8cm}
408 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
412 \begin{minipage}{4cm}
414 C-Si dimers (dumbbells)\\[-0.1cm]
415 on Si interstitial sites
419 \begin{minipage}{4.2cm}
421 Agglomeration of C-Si dumbbells\\[-0.1cm]
422 $\Rightarrow$ dark contrasts
426 \begin{minipage}{4cm}
428 Precipitation of 3C-SiC in Si\\[-0.1cm]
429 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
430 \& release of Si self-interstitials
434 \begin{minipage}{3.8cm}
436 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
440 \begin{minipage}{3.8cm}
442 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
446 \begin{minipage}{3.8cm}
448 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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464 Molecular dynamics (MD) simulations
473 \item Microscopic description of N particle system
474 \item Analytical interaction potential
475 \item Numerical integration using Newtons equation of motion\\
476 as a propagation rule in 6N-dimensional phase space
477 \item Observables obtained by time and/or ensemble averages
479 {\bf Details of the simulation:}
481 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
482 \item Ensemble: NpT (isothermal-isobaric)
484 \item Berendsen thermostat:
485 $\tau_{\text{T}}=100\text{ fs}$
486 \item Berendsen barostat:\\
487 $\tau_{\text{P}}=100\text{ fs}$,
488 $\beta^{-1}=100\text{ GPa}$
490 \item Erhart/Albe potential: Tersoff-like bond order potential
493 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
494 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
498 \begin{picture}(0,0)(-230,-30)
499 \includegraphics[width=5cm]{tersoff_angle.eps}
507 Density functional theory (DFT) calculations
512 Basic ingredients necessary for DFT
515 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
517 \item ... uniquely determines the ground state potential
519 \item ... minimizes the systems total energy
521 \item \underline{Born-Oppenheimer}
522 - $N$ moving electrons in an external potential of static nuclei
524 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
525 +\sum_i^N V_{\text{ext}}(r_i)
526 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
528 \item \underline{Effective potential}
529 - averaged electrostatic potential \& exchange and correlation
531 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
534 \item \underline{Kohn-Sham system}
535 - Schr\"odinger equation of N non-interacting particles
537 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
542 n(r)=\sum_i^N|\Phi_i(r)|^2
544 \item \underline{Self-consistent solution}\\
545 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
546 which in turn depends on $n(r)$
547 \item \underline{Variational principle}
548 - minimize total energy with respect to $n(r)$
556 Density functional theory (DFT) calculations
563 Details of applied DFT calculations in this work
566 \item \underline{Exchange correlation functional}
567 - approximations for the inhomogeneous electron gas
569 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
570 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
572 \item \underline{Plane wave basis set}
573 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
576 \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}}
578 \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
579 \item \underline{Pseudo potential}
580 - consider only the valence electrons
581 \item \underline{Code} - VASP 4.6
586 MD and structural optimization
589 \item MD integration: Gear predictor corrector algorithm
590 \item Pressure control: Parrinello-Rahman pressure control
591 \item Structural optimization: Conjugate gradient method
599 C and Si self-interstitial point defects in silicon
606 \begin{minipage}{8cm}
608 \begin{pspicture}(0,0)(7,5)
609 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
612 \item Creation of c-Si simulation volume
613 \item Periodic boundary conditions
614 \item $T=0\text{ K}$, $p=0\text{ bar}$
617 \rput(3.5,2.1){\rnode{insert}{\psframebox{
620 Insertion of interstitial C/Si atoms
623 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
626 Relaxation / structural energy minimization
629 \ncline[]{->}{init}{insert}
630 \ncline[]{->}{insert}{cool}
633 \begin{minipage}{5cm}
634 \includegraphics[width=5cm]{unit_cell_e.eps}\\
637 \begin{minipage}{9cm}
638 \begin{tabular}{l c c}
640 & size [unit cells] & \# atoms\\
642 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
643 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
647 \begin{minipage}{4cm}
648 {\color{red}$\bullet$} Tetrahedral\\
649 {\color{green}$\bullet$} Hexagonal\\
650 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
651 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
652 {\color{cyan}$\bullet$} Bond-centered\\
653 {\color{black}$\bullet$} Vacancy / Substitutional
662 \begin{minipage}{9.5cm}
665 Si self-interstitial point defects in silicon\\
668 \begin{tabular}{l c c c c c}
670 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
672 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
673 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
675 \end{tabular}\\[0.2cm]
677 \begin{minipage}{4.7cm}
678 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
680 \begin{minipage}{4.7cm}
682 {\tiny nearly T $\rightarrow$ T}\\
684 \includegraphics[width=4.7cm]{nhex_tet.ps}
687 \underline{Hexagonal} \hspace{2pt}
688 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
690 \begin{minipage}{2.7cm}
691 $E_{\text{f}}^*=4.48\text{ eV}$\\
692 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
694 \begin{minipage}{0.4cm}
699 \begin{minipage}{2.7cm}
700 $E_{\text{f}}=3.96\text{ eV}$\\
701 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
704 \begin{minipage}{2.9cm}
706 \underline{Vacancy}\\
707 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
712 \begin{minipage}{3.5cm}
715 \underline{\hkl<1 1 0> dumbbell}\\
716 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
717 \underline{Tetrahedral}\\
718 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
719 \underline{\hkl<1 0 0> dumbbell}\\
720 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
732 C interstitial point defects in silicon\\[-0.1cm]
735 \begin{tabular}{l c c c c c c}
737 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
739 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
740 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
742 \end{tabular}\\[0.1cm]
745 \begin{minipage}{2.7cm}
746 \underline{Hexagonal} \hspace{2pt}
747 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
748 $E_{\text{f}}^*=9.05\text{ eV}$\\
749 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
751 \begin{minipage}{0.4cm}
756 \begin{minipage}{2.7cm}
757 \underline{\hkl<1 0 0>}\\
758 $E_{\text{f}}=3.88\text{ eV}$\\
759 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
762 \begin{minipage}{2cm}
765 \begin{minipage}{3cm}
767 \underline{Tetrahedral}\\
768 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
773 \begin{minipage}{2.7cm}
774 \underline{Bond-centered}\\
775 $E_{\text{f}}^*=5.59\text{ eV}$\\
776 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
778 \begin{minipage}{0.4cm}
783 \begin{minipage}{2.7cm}
784 \underline{\hkl<1 1 0> dumbbell}\\
785 $E_{\text{f}}=5.18\text{ eV}$\\
786 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
789 \begin{minipage}{2cm}
792 \begin{minipage}{3cm}
794 \underline{Substitutional}\\
795 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
806 C \hkl<1 0 0> dumbbell interstitial configuration\\
810 \begin{tabular}{l c c c c c c c c}
812 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
814 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
815 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
817 \end{tabular}\\[0.2cm]
818 \begin{tabular}{l c c c c }
820 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
822 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
823 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
825 \end{tabular}\\[0.2cm]
826 \begin{tabular}{l c c c}
828 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
830 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
831 VASP & 0.109 & -0.065 & 0.174 \\
833 \end{tabular}\\[0.6cm]
836 \begin{minipage}{3.0cm}
838 \underline{Erhart/Albe}
839 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
842 \begin{minipage}{3.0cm}
845 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
849 \begin{picture}(0,0)(-185,10)
850 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
852 \begin{picture}(0,0)(-280,-150)
853 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
856 \begin{pspicture}(0,0)(0,0)
857 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
858 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
859 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
860 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
869 \begin{minipage}{8.5cm}
872 Bond-centered interstitial configuration\\[-0.1cm]
875 \begin{minipage}{3.0cm}
876 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
878 \begin{minipage}{5.2cm}
880 \item Linear Si-C-Si bond
881 \item Si: one C \& 3 Si neighbours
882 \item Spin polarized calculations
883 \item No saddle point!\\
890 \begin{minipage}[t]{6.5cm}
891 \begin{minipage}[t]{1.2cm}
893 {\tiny sp$^3$}\\[0.8cm]
894 \underline{${\color{black}\uparrow}$}
895 \underline{${\color{black}\uparrow}$}
896 \underline{${\color{black}\uparrow}$}
897 \underline{${\color{red}\uparrow}$}\\
900 \begin{minipage}[t]{1.4cm}
902 {\color{red}M}{\color{blue}O}\\[0.8cm]
903 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
904 $\sigma_{\text{ab}}$\\[0.5cm]
905 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
909 \begin{minipage}[t]{1.0cm}
913 \underline{${\color{white}\uparrow\uparrow}$}
914 \underline{${\color{white}\uparrow\uparrow}$}\\
916 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
917 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
921 \begin{minipage}[t]{1.4cm}
923 {\color{blue}M}{\color{green}O}\\[0.8cm]
924 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
925 $\sigma_{\text{ab}}$\\[0.5cm]
926 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
930 \begin{minipage}[t]{1.2cm}
933 {\tiny sp$^3$}\\[0.8cm]
934 \underline{${\color{green}\uparrow}$}
935 \underline{${\color{black}\uparrow}$}
936 \underline{${\color{black}\uparrow}$}
937 \underline{${\color{black}\uparrow}$}\\
945 \begin{minipage}{4.5cm}
946 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
948 \begin{minipage}{3.5cm}
949 {\color{gray}$\bullet$} Spin up\\
950 {\color{green}$\bullet$} Spin down\\
951 {\color{blue}$\bullet$} Resulting spin up\\
952 {\color{yellow}$\bullet$} Si atoms\\
953 {\color{red}$\bullet$} C atom
958 \begin{minipage}{4.2cm}
960 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
961 {\color{green}$\Box$} {\tiny unoccupied}\\
962 {\color{red}$\bullet$} {\tiny occupied}
971 Migration of the C \hkl<1 0 0> dumbbell interstitial
976 {\small Investigated pathways}
978 \begin{minipage}{8.5cm}
979 \begin{minipage}{8.3cm}
980 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
981 \begin{minipage}{2.4cm}
982 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
984 \begin{minipage}{0.4cm}
987 \begin{minipage}{2.4cm}
988 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
990 \begin{minipage}{0.4cm}
993 \begin{minipage}{2.4cm}
994 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
997 \begin{minipage}{8.3cm}
998 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
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/00-1-0-10_2333.eps}
1008 \begin{minipage}{0.4cm}
1011 \begin{minipage}{2.4cm}
1012 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1015 \begin{minipage}{8.3cm}
1016 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
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_ip0-10_2333.eps}
1026 \begin{minipage}{0.4cm}
1029 \begin{minipage}{2.4cm}
1030 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1035 \begin{minipage}{4.2cm}
1036 {\small Constrained relaxation\\
1037 technique (CRT) method}\\
1038 \includegraphics[width=4cm]{crt_orig.eps}
1040 \item Constrain diffusing atom
1041 \item Static constraints
1044 {\small Modifications}\\
1045 \includegraphics[width=4cm]{crt_mod.eps}
1047 \item Constrain all atoms
1048 \item Update individual\\
1059 Migration of the C \hkl<1 0 0> dumbbell interstitial
1065 \begin{minipage}{5.9cm}
1067 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
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1077 \includegraphics[width=1cm]{vasp_mig/bc.eps}
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1080 \includegraphics[width=1cm]{110_arrow.eps}
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1083 \includegraphics[height=0.9cm]{001_arrow.eps}
1089 \begin{minipage}{0.3cm}
1093 \begin{minipage}{5.9cm}
1095 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1098 \begin{picture}(0,0)(60,0)
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1110 \begin{picture}(0,0)(90,0)
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1121 \begin{minipage}{5.9cm}
1123 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1126 \begin{picture}(0,0)(60,0)
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1129 \begin{picture}(0,0)(10,0)
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1135 \begin{picture}(0,0)(12.5,10)
1136 \includegraphics[width=1cm]{100_arrow.eps}
1138 \begin{picture}(0,0)(90,0)
1139 \includegraphics[height=0.9cm]{001_arrow.eps}
1145 \begin{minipage}{0.3cm}
1148 \begin{minipage}{6.5cm}
1151 \item Energetically most favorable path
1154 \item Activation energy: $\approx$ 0.9 eV
1155 \item Experimental values: 0.73 ... 0.87 eV
1157 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1158 \item Reorientation (path 3)
1160 \item More likely composed of two consecutive steps of type 2
1161 \item Experimental values: 0.77 ... 0.88 eV
1163 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1172 Migration of the C \hkl<1 0 0> dumbbell interstitial
1177 \begin{minipage}{6.5cm}
1180 \begin{minipage}{5.9cm}
1182 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
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1186 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
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1189 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
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1195 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
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1198 \includegraphics[width=1cm]{110_arrow.eps}
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1216 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1218 \begin{picture}(0,0)(12.5,-5)
1219 \includegraphics[width=1cm]{100_arrow.eps}
1221 \begin{picture}(0,0)(90,-15)
1222 \includegraphics[height=0.9cm]{010_arrow.eps}
1228 \begin{minipage}{5.9cm}
1231 \item Lowest activation energy: $\approx$ 2.2 eV
1232 \item 2.4 times higher than VASP
1233 \item Different pathway
1234 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1239 \begin{minipage}{6.5cm}
1242 \begin{minipage}{5.9cm}
1244 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1247 \begin{pspicture}(0,0)(0,0)
1248 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1250 \begin{picture}(0,0)(60,-5)
1251 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1253 \begin{picture}(0,0)(0,-5)
1254 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1256 \begin{picture}(0,0)(-55,-5)
1257 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
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1260 \includegraphics[width=1cm]{100_arrow.eps}
1262 \begin{picture}(0,0)(90,0)
1263 \includegraphics[height=0.9cm]{001_arrow.eps}
1271 \begin{minipage}{5.9cm}
1272 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1283 Migrations involving the C \hkl<1 1 0> dumbbell interstitial
1292 \begin{minipage}{6.0cm}
1293 \includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
1295 \begin{minipage}{7cm}
1296 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1297 \hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
1298 \hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
1299 BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
1300 Composed of three single transitions\\[0.3cm]
1301 Activation energy of second transition slightly\\
1302 higher than direct transition (path 2)\\[0.3cm]
1303 $\Rightarrow$ very unlikely to happen
1304 \end{minipage}\\[0.2cm]
1308 \begin{minipage}{6.0cm}
1309 \includegraphics[width=6cm]{110_mig.ps}
1311 \begin{minipage}{7cm}
1312 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1313 \hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
1314 \hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
1315 \hkl<0 0 -1>\\[0.3cm]
1316 Composed of two single transitions\\[0.3cm]
1317 Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
1318 $\Rightarrow$ more readily constituting a probable transition
1326 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1336 E_{\text{f}}^{\text{defect combination}}-
1337 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1338 E_{\text{f}}^{\text{2nd defect}}
1344 \begin{tabular}{l c c c c c c}
1346 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1348 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1349 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1350 \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}\\
1351 \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}\\
1352 \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}\\
1353 \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}\\
1355 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1356 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1365 \begin{minipage}[t]{3.8cm}
1366 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1367 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1369 \begin{minipage}[t]{3.5cm}
1370 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1371 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1373 \begin{minipage}[t]{5.5cm}
1375 \item Restricted to VASP simulations
1376 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1377 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1378 \item Stress compensation / increase
1379 \item Most favorable: C clustering
1380 \item Unfavored: antiparallel orientations
1381 \item Indication of energetically favored\\
1386 \begin{picture}(0,0)(-295,-130)
1387 \includegraphics[width=3.5cm]{comb_pos.eps}
1395 Combinations of C-Si \hkl<1 0 0>-type interstitials
1402 Energetically most favorable combinations along \hkl<1 1 0>
1407 \begin{tabular}{l c c c c c c}
1409 & 1 & 2 & 3 & 4 & 5 & 6\\
1411 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1412 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1413 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>\\
1420 \begin{minipage}{7.0cm}
1421 \includegraphics[width=7cm]{db_along_110_cc.ps}
1423 \begin{minipage}{6.0cm}
1426 Interaction proportional to reciprocal cube of C-C distance
1428 Saturation in the immediate vicinity
1439 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1445 \begin{minipage}{3.2cm}
1446 \includegraphics[width=3cm]{sub_110_combo.eps}
1448 \begin{minipage}{7.8cm}
1449 \begin{tabular}{l c c c c c c}
1451 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1452 \hkl<1 0 1> & \hkl<-1 0 1> \\
1454 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1455 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1456 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1457 4 & \RM{4} & B & D & E & E & D \\
1458 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1465 \begin{tabular}{l c c c c c c c c c c}
1467 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1469 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1470 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1471 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1476 \begin{minipage}{6.0cm}
1477 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1479 \begin{minipage}{7cm}
1482 \item IBS: C may displace Si\\
1483 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1485 \hkl<1 1 0>-type $\rightarrow$ favored combination
1486 \renewcommand\labelitemi{$\Rightarrow$}
1487 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1488 ($E_{\text{f}}=3.88\text{ eV}$)
1489 \item Interaction drops quickly to zero\\
1490 (low interaction capture radius)
1499 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1506 \begin{minipage}[t]{3cm}
1507 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1508 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1510 \begin{minipage}[t]{7cm}
1513 Low activation energies\\
1514 High activation energies for reverse processes\\
1516 {\color{blue}C$_{\text{sub}}$ very stable}\\
1520 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1522 {\color{blue}Formation of SiC by successive substitution by C}
1526 \begin{minipage}[t]{3cm}
1527 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1528 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1533 \begin{minipage}{5.9cm}
1534 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1536 \begin{picture}(0,0)(70,0)
1537 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1539 \begin{picture}(0,0)(30,0)
1540 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1542 \begin{picture}(0,0)(-10,0)
1543 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1545 \begin{picture}(0,0)(-48,0)
1546 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1548 \begin{picture}(0,0)(12.5,5)
1549 \includegraphics[width=1cm]{100_arrow.eps}
1551 \begin{picture}(0,0)(97,-10)
1552 \includegraphics[height=0.9cm]{001_arrow.eps}
1558 \begin{minipage}{0.3cm}
1562 \begin{minipage}{5.9cm}
1563 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1565 \begin{picture}(0,0)(60,0)
1566 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1568 \begin{picture}(0,0)(25,0)
1569 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1571 \begin{picture}(0,0)(-20,0)
1572 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1574 \begin{picture}(0,0)(-55,0)
1575 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1577 \begin{picture}(0,0)(12.5,5)
1578 \includegraphics[width=1cm]{100_arrow.eps}
1580 \begin{picture}(0,0)(95,0)
1581 \includegraphics[height=0.9cm]{001_arrow.eps}
1593 Conclusion of defect / migration / combined defect simulations
1602 \item Accurately described by quantum-mechanical simulations
1603 \item Less correct description by classical potential simulations
1607 \item Consistent with solubility data of C in Si
1608 \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration
1609 \item Consistent with reorientation and diffusion experiments
1610 \item C migration pathway in Si identified
1615 Concerning the precipitation mechanism
1617 \item Agglomeration of C-Si dumbbells energetically favorable
1618 \item C-Si indeed favored compared to
1619 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1620 \item Possible low interaction capture radius of
1621 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1622 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1623 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1628 {\color{blue}Some results point to a different precipitation mechanism!}
1636 Silicon carbide precipitation simulations
1642 \begin{pspicture}(0,0)(12,6.5)
1644 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1647 \item Create c-Si volume
1648 \item Periodc boundary conditions
1649 \item Set requested $T$ and $p=0\text{ bar}$
1650 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1653 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1655 Insertion of C atoms at constant T
1657 \item total simulation volume {\pnode{in1}}
1658 \item volume of minimal SiC precipitate {\pnode{in2}}
1659 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1663 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1665 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1667 \ncline[]{->}{init}{insert}
1668 \ncline[]{->}{insert}{cool}
1669 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1670 \rput(7.8,6){\footnotesize $V_1$}
1671 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1672 \rput(9.2,4.85){\tiny $V_2$}
1673 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1674 \rput(9.55,4.45){\footnotesize $V_3$}
1675 \rput(7.9,3.2){\pnode{ins1}}
1676 \rput(9.22,2.8){\pnode{ins2}}
1677 \rput(11.0,2.4){\pnode{ins3}}
1678 \ncline[]{->}{in1}{ins1}
1679 \ncline[]{->}{in2}{ins2}
1680 \ncline[]{->}{in3}{ins3}
1685 \item Restricted to classical potential simulations
1686 \item $V_2$ and $V_3$ considered due to low diffusion
1687 \item Amount of C atoms: 6000
1688 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1689 \item Simulation volume: $31\times 31\times 31$ unit cells
1698 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1703 \begin{minipage}{6.5cm}
1704 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1706 \begin{minipage}{6.5cm}
1707 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1710 \begin{minipage}{6.5cm}
1711 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1713 \begin{minipage}{6.5cm}
1715 \underline{Low C concentration ($V_1$)}\\
1716 \hkl<1 0 0> C-Si dumbbell dominated structure
1718 \item Si-C bumbs around 0.19 nm
1719 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1720 concatenated dumbbells of various orientation
1721 \item Si-Si NN distance stretched to 0.3 nm
1723 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1724 \underline{High C concentration ($V_2$, $V_3$)}\\
1725 High amount of strongly bound C-C bonds\\
1726 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1727 Only short range order observable\\
1728 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1736 Limitations of molecular dynamics and short range potentials
1743 \underline{Time scale problem of MD}\\[0.2cm]
1744 Minimize integration error\\
1745 $\Rightarrow$ discretization considerably smaller than
1746 reciprocal of fastest vibrational mode\\[0.1cm]
1747 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1748 $\Rightarrow$ suitable choice of time step:
1749 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1750 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1751 Several local minima in energy surface separated by large energy barriers\\
1752 $\Rightarrow$ transition event corresponds to a multiple
1753 of vibrational periods\\
1754 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1755 infrequent transition events\\[0.1cm]
1756 {\color{blue}Accelerated methods:}
1757 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1761 \underline{Limitations related to the short range potential}\\[0.2cm]
1762 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1763 and 2$^{\text{nd}}$ next neighbours\\
1764 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1770 Potential enhanced problem of slow phase space propagation
1775 \underline{Approach to the (twofold) problem}\\[0.2cm]
1776 Increased temperature simulations without TAD corrections\\
1777 (accelerated methods or higher time scales exclusively not sufficient)
1779 \begin{picture}(0,0)(-260,-30)
1781 \begin{minipage}{4.2cm}
1788 \item 3C-SiC also observed for higher T
1789 \item higher T inside sample
1790 \item structural evolution vs.\\
1791 equilibrium properties
1797 \begin{picture}(0,0)(-305,-155)
1799 \begin{minipage}{2.5cm}
1803 thermodynmic sampling
1814 Increased temperature simulations at low C concentration
1819 \begin{minipage}{6.5cm}
1820 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1822 \begin{minipage}{6.5cm}
1823 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1826 \begin{minipage}{6.5cm}
1827 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1829 \begin{minipage}{6.5cm}
1831 \underline{Si-C bonds:}
1833 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1834 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1836 \underline{Si-Si bonds:}
1837 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1838 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1839 \underline{C-C bonds:}
1841 \item C-C next neighbour pairs reduced (mandatory)
1842 \item Peak at 0.3 nm slightly shifted
1844 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1845 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1847 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1849 \item Range [|-$\downarrow$]:
1850 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1851 with nearby Si$_{\text{I}}$}
1856 \begin{picture}(0,0)(-330,-74)
1859 \begin{minipage}{1.6cm}
1862 stretched SiC\\[-0.1cm]
1874 Increased temperature simulations at high C concentration
1879 \begin{minipage}{6.5cm}
1880 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1882 \begin{minipage}{6.5cm}
1883 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1887 Decreasing cut-off artifact\\
1888 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1889 $\Rightarrow$ hard to categorize
1895 \begin{minipage}[t]{6.0cm}
1896 0.186 nm: Si-C pairs $\uparrow$\\
1897 (as expected in 3C-SiC)\\[0.2cm]
1898 0.282 nm: Si-C-C\\[0.2cm]
1899 $\approx$0.35 nm: C-Si-Si
1902 \begin{minipage}{0.2cm}
1906 \begin{minipage}[t]{6.0cm}
1907 0.15 nm: C-C pairs $\uparrow$\\
1908 (as expected in graphite/diamond)\\[0.2cm]
1909 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1910 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1917 {\color{red}Amorphous} SiC-like phase remains\\
1918 Slightly sharper peaks
1919 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
1920 due to temperature\\[0.1cm]
1923 Continue with higher temperatures and longer time scales
1932 Valuation of a practicable temperature limit
1942 Recrystallization is a hard task!
1943 $\Rightarrow$ Avoid melting!
1952 \begin{minipage}{7.5cm}
1953 \includegraphics[width=7cm]{fe_and_t.ps}
1955 \begin{minipage}{5.5cm}
1956 \underline{Melting does not occur instantly after}\\
1957 \underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
1959 \item required transition enthalpy
1960 \item hysterisis behaviour
1962 \underline{Heating up c-Si by 1 K/ps}
1964 \item transition occurs at $\approx$ 3125 K
1965 \item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
1966 (literature: 50.2 kJ/mole)
1973 \begin{minipage}{4cm}
1974 Initially chosen temperatures:\\
1975 $1.0 - 1.2 \cdot T_{\text{m}}$
1978 \begin{minipage}{3cm}
1984 \begin{minipage}{5cm}
1985 Introduced C (defects)\\
1986 $\rightarrow$ reduction of transition point\\
1987 $\rightarrow$ melting already at $T_{\text{m}}$
1996 Maximum temperature used: $0.95\cdot T_{\text{m}}$
2006 Long time scale simulations at maximum temperature
2013 \underline{Differences}
2015 \item Cubic volume $\Rightarrow$ spherical volume
2016 \item Amount of C atoms: 6000 $\rightarrow$ 5500
2017 \item Temperature set to $0.95 \cdot T_{\text{m}}$
2018 \item Simulation volume: 21 unit cells of c-Si in each direction
2025 Simulations in progress! :)\\
2027 ... show evolution of radial distribution in ns timesteps ...
2037 Investigation of a silicon carbide precipitate in silicon
2046 \begin{minipage}{5.3cm}
2048 \frac{8}{a_{\text{Si}}^3}(
2049 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
2050 -\frac{4}{3}\pi x^3)+
2051 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
2058 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
2059 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
2062 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
2066 \begin{minipage}{0.3cm}
2069 \begin{minipage}{7.0cm}
2070 \underline{Construction}
2072 \item Simulation volume: 21$^3$ unit cells of c-Si
2073 \item Spherical topotactically aligned precipitate\\
2074 $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
2075 \item Create c-Si but skipped inside sphere of radius $x$
2076 \item Create 3C-SiC inside sphere of radius $x$\\
2077 and lattice constant $y$
2078 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
2084 \begin{minipage}{6.2cm}
2085 \includegraphics[width=6cm,draft=false]{pc_0.ps}
2087 \begin{minipage}{6.8cm}
2090 \item Slight increase of c-Si lattice constant!
2091 \item C-C peaks (imply same distanced Si-Si peaks)
2093 \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
2094 \item Bumps ({\color{green}$\downarrow$}):
2095 4$^{\text{th}}$ and 6$^{\text{th}}$ NN
2097 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
2098 $\rightarrow$ compressed precipitate
2099 \item Interface tension:\\
2100 20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
2101 (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
2110 Investigation of a silicon carbide precipitate in silicon
2115 \begin{minipage}{7cm}
2116 \underline{Appended annealing steps}
2118 \item artificially constructed interface\\
2119 $\rightarrow$ allow for rearrangement of interface atoms
2120 \item check SiC stability
2122 \underline{Temperature schedule}
2124 \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
2126 \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
2128 $\rightarrow$ melting at around 2840 K
2129 (\href{../video/sic_prec_120.avi}{$\rhd$})
2130 \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
2131 $\rightarrow$ no energetically more favorable struture
2134 \begin{minipage}{6cm}
2135 \includegraphics[width=6.7cm]{fe_and_t_sic.ps}
2138 \begin{minipage}{4cm}
2139 \includegraphics[width=4cm]{sic_prec/melt_01.eps}
2141 \begin{minipage}{0.4cm}
2144 \begin{minipage}{4cm}
2145 \includegraphics[width=4cm]{sic_prec/melt_02.eps}
2147 \begin{minipage}{0.4cm}
2150 \begin{minipage}{4cm}
2151 \includegraphics[width=4cm]{sic_prec/melt_03.eps}
2159 Summary / Conclusion / Outlook
2167 \begin{minipage}{12.9cm}
2170 \item Summary \& conclusion
2172 \item Point defects excellently / fairly well described
2173 by QM / classical potential simulations
2174 \item Identified migration path explaining
2175 diffusion and reorientation experiments
2176 \item Agglomeration of point defects energetically favorable
2177 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2181 \item Discussions concerning interpretation of QM results (Paderborn)
2182 \item Compare migration barrier of
2183 \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell
2184 \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \&
2185 C-Si \hkl<1 0 0> dumbbell (IBS)
2194 \begin{minipage}[t]{6.2cm}
2195 \underline{Pecipitation simulations}
2197 \item Summary \& conclusion
2200 $\rightarrow$ C-Si \hkl<1 0 0> dumbbell\\
2202 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2203 \item High C concentration\\
2204 $\rightarrow$ amorphous SiC like phase
2208 \item Accelerated method: self-guided MD
2209 \item Activation relaxation technique
2210 \item Constrainted transition path
2216 \begin{minipage}[t]{6.2cm}
2217 \underline{Constructed 3C-SiC precipitate}
2219 \item Summary \& conclusion
2221 \item Small / stable / compressed 3C-SiC\\
2222 precipitate in slightly stretched\\
2224 \item Interface tension matches experiemnts
2228 \item Try to improve interface
2229 \item Precipitates of different size