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89 Atomistic simulation study of the silicon carbide precipitation
95 \textsc{F. Zirkelbach}
108 % motivation / properties / applications of silicon carbide
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122 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
124 \rput[lt](0.5,4){wide band gap}
125 \rput[lt](0.5,3.5){high electric breakdown field}
126 \rput[lt](0.5,3){good electron mobility}
127 \rput[lt](0.5,2.5){high electron saturation drift velocity}
128 \rput[lt](0.5,2){high thermal conductivity}
130 \rput[lt](0.5,1.5){hard and mechanically stable}
131 \rput[lt](0.5,1){chemically inert}
133 \rput[lt](0.5,0.5){radiation hardness}
135 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
137 \rput[rt](13,3.85){high-temperature, high power}
138 \rput[rt](13,3.5){and high-frequency}
139 \rput[rt](13,3.15){electronic and optoelectronic devices}
141 \rput[rt](13,2.35){material suitable for extreme conditions}
142 \rput[rt](13,2){microelectromechanical systems}
143 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
145 \rput[rt](13,0.85){first wall reactor material, detectors}
146 \rput[rt](13,0.5){and electronic devices for space}
150 \begin{picture}(0,0)(-10,68)
151 \includegraphics[width=2.6cm]{wide_band_gap.eps}
153 \begin{picture}(0,0)(-295,-165)
154 \includegraphics[width=3cm]{sic_led.eps}
156 \begin{picture}(0,0)(-215,-165)
157 \includegraphics[width=2.5cm]{6h-sic_3c-sic.eps}
159 \begin{picture}(0,0)(-313,65)
160 \includegraphics[width=2.2cm]{infineon_schottky.eps}
162 \begin{picture}(0,0)(-220,65)
163 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
177 \item Polyteps and fabrication of silicon carbide
178 \item Supposed precipitation mechanism of SiC in Si
179 \item Utilized simulation techniques
181 \item Molecular dynamics (MD) simulations
182 \item Density functional theory (DFT) calculations
184 \item C and Si self-interstitial point defects in silicon
185 \item Silicon carbide precipitation simulations
186 \item Investigation of a silicon carbide precipitate in silicon
187 \item Summary / Conclusion / Outlook
204 \begin{tabular}{l c c c c c c}
206 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
208 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
209 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
210 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
211 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
212 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
213 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
214 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
222 \begin{picture}(0,0)(-160,-155)
223 \includegraphics[width=7cm]{polytypes.eps}
225 \begin{picture}(0,0)(-10,-185)
226 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
228 \begin{picture}(0,0)(-10,-175)
229 {\tiny cubic (twist)}
231 \begin{picture}(0,0)(-60,-175)
232 {\tiny hexagonal (no twist)}
234 \begin{pspicture}(0,0)(0,0)
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240 \begin{pspicture}(0,0)(0,0)
241 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
249 Fabrication of silicon carbide
256 SiC - \emph{Born from the stars, perfected on earth.}
260 Conventional thin film SiC growth:
262 \item \underline{Sublimation growth using the modified Lely method}
264 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
265 \item Surrounded by polycrystalline SiC in a graphite crucible\\
266 at $T=2100-2400 \, ^{\circ} \text{C}$
267 \item Deposition of supersaturated vapor on cooler seed crystal
269 \item \underline{Homoepitaxial growth using CVD}
271 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
272 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
273 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
274 \item High quality but limited in size of substrates
276 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
278 \item Two steps: carbonization and growth
279 \item $T=650-1050 \, ^{\circ} \text{C}$
280 \item Quality and size not yet sufficient
284 \begin{picture}(0,0)(-280,-65)
285 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
287 \begin{picture}(0,0)(-280,-55)
288 \begin{minipage}{5cm}
290 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
295 \begin{picture}(0,0)(-265,-150)
296 \includegraphics[width=2.4cm]{m_lely.eps}
298 \begin{picture}(0,0)(-333,-175)
299 \begin{minipage}{5cm}
305 5. Insulation\\[-7pt]
316 Fabrication of silicon carbide
321 Alternative approach:
322 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
324 \item \underline{Implantation step 1}\\
325 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
326 $\Rightarrow$ box-like distribution of equally sized
327 and epitactically oriented SiC precipitates
329 \item \underline{Implantation step 2}\\
330 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
331 $\Rightarrow$ destruction of SiC nanocrystals
332 in growing amorphous interface layers
333 \item \underline{Annealing}\\
334 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
335 $\Rightarrow$ homogeneous, stoichiometric SiC layer
336 with sharp interfaces
339 \begin{minipage}{6.3cm}
340 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
342 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
346 \begin{minipage}{6.3cm}
349 Precipitation mechanism not yet fully understood!
351 \renewcommand\labelitemi{$\Rightarrow$}
353 \underline{Understanding the SiC precipitation}
355 \item significant technological progress in SiC thin film formation
356 \item perspectives for processes relying upon prevention of SiC precipitation
367 Supposed precipitation mechanism of SiC in Si
374 \begin{minipage}{3.8cm}
375 Si \& SiC lattice structure\\[0.2cm]
376 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
380 \begin{minipage}{3.8cm}
382 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
386 \begin{minipage}{3.8cm}
388 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
392 \begin{minipage}{4cm}
394 C-Si dimers (dumbbells)\\[-0.1cm]
395 on Si interstitial sites
399 \begin{minipage}{4.2cm}
401 Agglomeration of C-Si dumbbells\\[-0.1cm]
402 $\Rightarrow$ dark contrasts
406 \begin{minipage}{4cm}
408 Precipitation of 3C-SiC in Si\\[-0.1cm]
409 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
410 \& release of Si self-interstitials
414 \begin{minipage}{3.8cm}
416 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
420 \begin{minipage}{3.8cm}
422 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
426 \begin{minipage}{3.8cm}
428 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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444 Molecular dynamics (MD) simulations
453 \item Microscopic description of N particle system
454 \item Analytical interaction potential
455 \item Numerical integration using Newtons equation of motion\\
456 as a propagation rule in 6N-dimensional phase space
457 \item Observables obtained by time and/or ensemble averages
459 {\bf Details of the simulation:}
461 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
462 \item Ensemble: NpT (isothermal-isobaric)
464 \item Berendsen thermostat:
465 $\tau_{\text{T}}=100\text{ fs}$
466 \item Berendsen barostat:\\
467 $\tau_{\text{P}}=100\text{ fs}$,
468 $\beta^{-1}=100\text{ GPa}$
470 \item Erhart/Albe potential: Tersoff-like bond order potential
473 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
474 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
478 \begin{picture}(0,0)(-230,-30)
479 \includegraphics[width=5cm]{tersoff_angle.eps}
487 Density functional theory (DFT) calculations
492 Basic ingredients necessary for DFT
495 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
497 \item ... uniquely determines the ground state potential
499 \item ... minimizes the systems total energy
501 \item \underline{Born-Oppenheimer}
502 - $N$ moving electrons in an external potential of static nuclei
504 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
505 +\sum_i^N V_{\text{ext}}(r_i)
506 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
508 \item \underline{Effective potential}
509 - averaged electrostatic potential \& exchange and correlation
511 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
514 \item \underline{Kohn-Sham system}
515 - Schr\"odinger equation of N non-interacting particles
517 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
522 n(r)=\sum_i^N|\Phi_i(r)|^2
524 \item \underline{Self-consistent solution}\\
525 $n(r)$ depends on $\Phi_i$, which depends on $V_{\text{eff}}$,
526 which in turn depends on $n(r)$
527 \item \underline{Variational principle}
528 - minimize total energy with respect to $n(r)$
536 Density functional theory (DFT) calculations
543 Details of applied DFT calculations in this work
546 \item \underline{Exchange correlation functional}
547 - approximations for the inhomogeneous electron gas
549 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
550 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
552 \item \underline{Plane wave basis set}
553 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
556 \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}}
558 \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
559 \item \underline{Pseudo potential}
560 - consider only the valence electrons
561 \item \underline{Code} - VASP 4.6
566 MD and structural optimization
569 \item MD integration: Gear predictor corrector algorithm
570 \item Pressure control: Parrinello-Rahman pressure control
571 \item Structural optimization: Conjugate gradient method
579 C and Si self-interstitial point defects in silicon
586 \begin{minipage}{8cm}
588 \begin{pspicture}(0,0)(7,5)
589 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
592 \item Creation of c-Si simulation volume
593 \item Periodic boundary conditions
594 \item $T=0\text{ K}$, $p=0\text{ bar}$
597 \rput(3.5,2.1){\rnode{insert}{\psframebox{
600 Insertion of interstitial C/Si atoms
603 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
606 Relaxation / structural energy minimization
609 \ncline[]{->}{init}{insert}
610 \ncline[]{->}{insert}{cool}
613 \begin{minipage}{5cm}
614 \includegraphics[width=5cm]{unit_cell_e.eps}\\
617 \begin{minipage}{9cm}
618 \begin{tabular}{l c c}
620 & size [unit cells] & \# atoms\\
622 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
623 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
627 \begin{minipage}{4cm}
628 {\color{red}$\bullet$} Tetrahedral\\
629 {\color{green}$\bullet$} Hexagonal\\
630 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
631 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
632 {\color{cyan}$\bullet$} Bond-centered\\
633 {\color{black}$\bullet$} Vacancy / Substitutional
642 \begin{minipage}{9.5cm}
645 Si self-interstitial point defects in silicon\\
648 \begin{tabular}{l c c c c c}
650 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
652 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
653 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
655 \end{tabular}\\[0.2cm]
657 \begin{minipage}{4.7cm}
658 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
660 \begin{minipage}{4.7cm}
662 {\tiny nearly T $\rightarrow$ T}\\
664 \includegraphics[width=4.7cm]{nhex_tet.ps}
667 \underline{Hexagonal} \hspace{2pt}
668 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
670 \begin{minipage}{2.7cm}
671 $E_{\text{f}}^*=4.48\text{ eV}$\\
672 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
674 \begin{minipage}{0.4cm}
679 \begin{minipage}{2.7cm}
680 $E_{\text{f}}=3.96\text{ eV}$\\
681 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
684 \begin{minipage}{2.9cm}
686 \underline{Vacancy}\\
687 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
692 \begin{minipage}{3.5cm}
695 \underline{\hkl<1 1 0> dumbbell}\\
696 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
697 \underline{Tetrahedral}\\
698 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
699 \underline{\hkl<1 0 0> dumbbell}\\
700 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
712 C interstitial point defects in silicon\\[-0.1cm]
715 \begin{tabular}{l c c c c c c}
717 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
719 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
720 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
722 \end{tabular}\\[0.1cm]
725 \begin{minipage}{2.7cm}
726 \underline{Hexagonal} \hspace{2pt}
727 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
728 $E_{\text{f}}^*=9.05\text{ eV}$\\
729 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
731 \begin{minipage}{0.4cm}
736 \begin{minipage}{2.7cm}
737 \underline{\hkl<1 0 0>}\\
738 $E_{\text{f}}=3.88\text{ eV}$\\
739 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
742 \begin{minipage}{2cm}
745 \begin{minipage}{3cm}
747 \underline{Tetrahedral}\\
748 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
753 \begin{minipage}{2.7cm}
754 \underline{Bond-centered}\\
755 $E_{\text{f}}^*=5.59\text{ eV}$\\
756 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
758 \begin{minipage}{0.4cm}
763 \begin{minipage}{2.7cm}
764 \underline{\hkl<1 1 0> dumbbell}\\
765 $E_{\text{f}}=5.18\text{ eV}$\\
766 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
769 \begin{minipage}{2cm}
772 \begin{minipage}{3cm}
774 \underline{Substitutional}\\
775 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
786 C \hkl<1 0 0> dumbbell interstitial configuration\\
790 \begin{tabular}{l c c c c c c c c}
792 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
794 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
795 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
797 \end{tabular}\\[0.2cm]
798 \begin{tabular}{l c c c c }
800 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
802 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
803 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
805 \end{tabular}\\[0.2cm]
806 \begin{tabular}{l c c c}
808 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
810 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
811 VASP & 0.109 & -0.065 & 0.174 \\
813 \end{tabular}\\[0.6cm]
816 \begin{minipage}{3.0cm}
818 \underline{Erhart/Albe}
819 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
822 \begin{minipage}{3.0cm}
825 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
829 \begin{picture}(0,0)(-185,10)
830 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
832 \begin{picture}(0,0)(-280,-150)
833 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
836 \begin{pspicture}(0,0)(0,0)
837 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
838 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
839 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
840 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
849 \begin{minipage}{8.5cm}
852 Bond-centered interstitial configuration\\[-0.1cm]
855 \begin{minipage}{3.0cm}
856 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
858 \begin{minipage}{5.2cm}
860 \item Linear Si-C-Si bond
861 \item Si: one C \& 3 Si neighbours
862 \item Spin polarized calculations
863 \item No saddle point!\\
870 \begin{minipage}[t]{6.5cm}
871 \begin{minipage}[t]{1.2cm}
873 {\tiny sp$^3$}\\[0.8cm]
874 \underline{${\color{black}\uparrow}$}
875 \underline{${\color{black}\uparrow}$}
876 \underline{${\color{black}\uparrow}$}
877 \underline{${\color{red}\uparrow}$}\\
880 \begin{minipage}[t]{1.4cm}
882 {\color{red}M}{\color{blue}O}\\[0.8cm]
883 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
884 $\sigma_{\text{ab}}$\\[0.5cm]
885 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
889 \begin{minipage}[t]{1.0cm}
893 \underline{${\color{white}\uparrow\uparrow}$}
894 \underline{${\color{white}\uparrow\uparrow}$}\\
896 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
897 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
901 \begin{minipage}[t]{1.4cm}
903 {\color{blue}M}{\color{green}O}\\[0.8cm]
904 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
905 $\sigma_{\text{ab}}$\\[0.5cm]
906 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
910 \begin{minipage}[t]{1.2cm}
913 {\tiny sp$^3$}\\[0.8cm]
914 \underline{${\color{green}\uparrow}$}
915 \underline{${\color{black}\uparrow}$}
916 \underline{${\color{black}\uparrow}$}
917 \underline{${\color{black}\uparrow}$}\\
925 \begin{minipage}{4.5cm}
926 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
928 \begin{minipage}{3.5cm}
929 {\color{gray}$\bullet$} Spin up\\
930 {\color{green}$\bullet$} Spin down\\
931 {\color{blue}$\bullet$} Resulting spin up\\
932 {\color{yellow}$\bullet$} Si atoms\\
933 {\color{red}$\bullet$} C atom
938 \begin{minipage}{4.2cm}
940 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
941 {\color{green}$\Box$} {\tiny unoccupied}\\
942 {\color{red}$\bullet$} {\tiny occupied}
951 Migration of the C \hkl<1 0 0> dumbbell interstitial
956 {\small Investigated pathways}
958 \begin{minipage}{8.5cm}
959 \begin{minipage}{8.3cm}
960 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
961 \begin{minipage}{2.4cm}
962 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
964 \begin{minipage}{0.4cm}
967 \begin{minipage}{2.4cm}
968 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
970 \begin{minipage}{0.4cm}
973 \begin{minipage}{2.4cm}
974 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
977 \begin{minipage}{8.3cm}
978 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
979 \begin{minipage}{2.4cm}
980 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
982 \begin{minipage}{0.4cm}
985 \begin{minipage}{2.4cm}
986 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
988 \begin{minipage}{0.4cm}
991 \begin{minipage}{2.4cm}
992 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
995 \begin{minipage}{8.3cm}
996 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
997 \begin{minipage}{2.4cm}
998 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1000 \begin{minipage}{0.4cm}
1003 \begin{minipage}{2.4cm}
1004 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1006 \begin{minipage}{0.4cm}
1009 \begin{minipage}{2.4cm}
1010 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1015 \begin{minipage}{4.2cm}
1016 {\small Constrained relaxation\\
1017 technique (CRT) method}\\
1018 \includegraphics[width=4cm]{crt_orig.eps}
1020 \item Constrain diffusing atom
1021 \item Static constraints
1024 {\small Modifications}\\
1025 \includegraphics[width=4cm]{crt_mod.eps}
1027 \item Constrain all atoms
1028 \item Update individual\\
1039 Migration of the C \hkl<1 0 0> dumbbell interstitial
1045 \begin{minipage}{5.9cm}
1047 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
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1054 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1056 \begin{picture}(0,0)(-55,0)
1057 \includegraphics[width=1cm]{vasp_mig/bc.eps}
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1060 \includegraphics[width=1cm]{110_arrow.eps}
1062 \begin{picture}(0,0)(90,0)
1063 \includegraphics[height=0.9cm]{001_arrow.eps}
1069 \begin{minipage}{0.3cm}
1073 \begin{minipage}{5.9cm}
1075 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1078 \begin{picture}(0,0)(60,0)
1079 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1081 \begin{picture}(0,0)(5,0)
1082 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
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1088 \includegraphics[width=1cm]{100_arrow.eps}
1090 \begin{picture}(0,0)(90,0)
1091 \includegraphics[height=0.9cm]{001_arrow.eps}
1101 \begin{minipage}{5.9cm}
1103 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1106 \begin{picture}(0,0)(60,0)
1107 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1109 \begin{picture}(0,0)(10,0)
1110 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1112 \begin{picture}(0,0)(-60,0)
1113 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1115 \begin{picture}(0,0)(12.5,10)
1116 \includegraphics[width=1cm]{100_arrow.eps}
1118 \begin{picture}(0,0)(90,0)
1119 \includegraphics[height=0.9cm]{001_arrow.eps}
1125 \begin{minipage}{0.3cm}
1128 \begin{minipage}{6.5cm}
1131 \item Energetically most favorable path
1134 \item Activation energy: $\approx$ 0.9 eV
1135 \item Experimental values: 0.73 ... 0.87 eV
1137 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1138 \item Reorientation (path 3)
1140 \item More likely composed of two consecutive steps of type 2
1141 \item Experimental values: 0.77 ... 0.88 eV
1143 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1152 Migration of the C \hkl<1 0 0> dumbbell interstitial
1157 \begin{minipage}{6.5cm}
1160 \begin{minipage}{5.9cm}
1162 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1165 \begin{pspicture}(0,0)(0,0)
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1168 \begin{picture}(0,0)(60,-50)
1169 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1171 \begin{picture}(0,0)(5,-50)
1172 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1174 \begin{picture}(0,0)(-55,-50)
1175 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1177 \begin{picture}(0,0)(12.5,-40)
1178 \includegraphics[width=1cm]{110_arrow.eps}
1180 \begin{picture}(0,0)(90,-45)
1181 \includegraphics[height=0.9cm]{001_arrow.eps}
1183 \begin{pspicture}(0,0)(0,0)
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1187 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1189 \begin{picture}(0,0)(35,-15)
1190 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1192 \begin{picture}(0,0)(-5,-15)
1193 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1195 \begin{picture}(0,0)(-55,-15)
1196 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1198 \begin{picture}(0,0)(12.5,-5)
1199 \includegraphics[width=1cm]{100_arrow.eps}
1201 \begin{picture}(0,0)(90,-15)
1202 \includegraphics[height=0.9cm]{010_arrow.eps}
1208 \begin{minipage}{5.9cm}
1211 \item Lowest activation energy: $\approx$ 2.2 eV
1212 \item 2.4 times higher than VASP
1213 \item Different pathway
1214 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1219 \begin{minipage}{6.5cm}
1222 \begin{minipage}{5.9cm}
1224 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1227 \begin{pspicture}(0,0)(0,0)
1228 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1230 \begin{picture}(0,0)(60,-5)
1231 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1233 \begin{picture}(0,0)(0,-5)
1234 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1236 \begin{picture}(0,0)(-55,-5)
1237 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1239 \begin{picture}(0,0)(12.5,5)
1240 \includegraphics[width=1cm]{100_arrow.eps}
1242 \begin{picture}(0,0)(90,0)
1243 \includegraphics[height=0.9cm]{001_arrow.eps}
1251 \begin{minipage}{5.9cm}
1252 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1263 Migration involving the C \hkl<1 1 0> dumbbell interstitial
1274 Combinations of point defects
1285 Silicon carbide precipitation simulations
1295 Investigation of a silicon carbide precipitate in silicon