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92 Atomistic simulation study of the silicon carbide precipitation
98 \textsc{F. Zirkelbach}
111 % motivation / properties / applications of silicon carbide
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125 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
127 \rput[lt](0.5,4){wide band gap}
128 \rput[lt](0.5,3.5){high electric breakdown field}
129 \rput[lt](0.5,3){good electron mobility}
130 \rput[lt](0.5,2.5){high electron saturation drift velocity}
131 \rput[lt](0.5,2){high thermal conductivity}
133 \rput[lt](0.5,1.5){hard and mechanically stable}
134 \rput[lt](0.5,1){chemically inert}
136 \rput[lt](0.5,0.5){radiation hardness}
138 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
140 \rput[rt](13,3.85){high-temperature, high power}
141 \rput[rt](13,3.5){and high-frequency}
142 \rput[rt](13,3.15){electronic and optoelectronic devices}
144 \rput[rt](13,2.35){material suitable for extreme conditions}
145 \rput[rt](13,2){microelectromechanical systems}
146 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
148 \rput[rt](13,0.85){first wall reactor material, detectors}
149 \rput[rt](13,0.5){and electronic devices for space}
153 \begin{picture}(0,0)(-10,68)
154 \includegraphics[width=2.6cm]{wide_band_gap.eps}
156 \begin{picture}(0,0)(-295,-165)
157 \includegraphics[width=3cm]{sic_led.eps}
159 \begin{picture}(0,0)(-215,-165)
160 \includegraphics[width=2.5cm]{6h-sic_3c-sic.eps}
162 \begin{picture}(0,0)(-313,65)
163 \includegraphics[width=2.2cm]{infineon_schottky.eps}
165 \begin{picture}(0,0)(-220,65)
166 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
180 \item Polyteps and fabrication of silicon carbide
181 \item Supposed precipitation mechanism of SiC in Si
182 \item Utilized simulation techniques
184 \item Molecular dynamics (MD) simulations
185 \item Density functional theory (DFT) calculations
187 \item C and Si self-interstitial point defects in silicon
188 \item Silicon carbide precipitation simulations
189 \item Investigation of a silicon carbide precipitate in silicon
190 \item Summary / Conclusion / Outlook
207 \begin{tabular}{l c c c c c c}
209 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
211 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
212 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
213 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
214 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
215 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
216 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
217 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
225 \begin{picture}(0,0)(-160,-155)
226 \includegraphics[width=7cm]{polytypes.eps}
228 \begin{picture}(0,0)(-10,-185)
229 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
231 \begin{picture}(0,0)(-10,-175)
232 {\tiny cubic (twist)}
234 \begin{picture}(0,0)(-60,-175)
235 {\tiny hexagonal (no twist)}
237 \begin{pspicture}(0,0)(0,0)
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243 \begin{pspicture}(0,0)(0,0)
244 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
252 Fabrication of silicon carbide
259 SiC - \emph{Born from the stars, perfected on earth.}
263 Conventional thin film SiC growth:
265 \item \underline{Sublimation growth using the modified Lely method}
267 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
268 \item Surrounded by polycrystalline SiC in a graphite crucible\\
269 at $T=2100-2400 \, ^{\circ} \text{C}$
270 \item Deposition of supersaturated vapor on cooler seed crystal
272 \item \underline{Homoepitaxial growth using CVD}
274 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
275 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
276 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
277 \item High quality but limited in size of substrates
279 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
281 \item Two steps: carbonization and growth
282 \item $T=650-1050 \, ^{\circ} \text{C}$
283 \item Quality and size not yet sufficient
287 \begin{picture}(0,0)(-280,-65)
288 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
290 \begin{picture}(0,0)(-280,-55)
291 \begin{minipage}{5cm}
293 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
298 \begin{picture}(0,0)(-265,-150)
299 \includegraphics[width=2.4cm]{m_lely.eps}
301 \begin{picture}(0,0)(-333,-175)
302 \begin{minipage}{5cm}
308 5. Insulation\\[-7pt]
319 Fabrication of silicon carbide
324 Alternative approach:
325 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
327 \item \underline{Implantation step 1}\\
328 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
329 $\Rightarrow$ box-like distribution of equally sized
330 and epitactically oriented SiC precipitates
332 \item \underline{Implantation step 2}\\
333 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
334 $\Rightarrow$ destruction of SiC nanocrystals
335 in growing amorphous interface layers
336 \item \underline{Annealing}\\
337 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
338 $\Rightarrow$ homogeneous, stoichiometric SiC layer
339 with sharp interfaces
342 \begin{minipage}{6.3cm}
343 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
345 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
349 \begin{minipage}{6.3cm}
352 Precipitation mechanism not yet fully understood!
354 \renewcommand\labelitemi{$\Rightarrow$}
356 \underline{Understanding the SiC precipitation}
358 \item significant technological progress in SiC thin film formation
359 \item perspectives for processes relying upon prevention of SiC precipitation
370 Supposed precipitation mechanism of SiC in Si
377 \begin{minipage}{3.8cm}
378 Si \& SiC lattice structure\\[0.2cm]
379 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
383 \begin{minipage}{3.8cm}
385 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
389 \begin{minipage}{3.8cm}
391 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
395 \begin{minipage}{4cm}
397 C-Si dimers (dumbbells)\\[-0.1cm]
398 on Si interstitial sites
402 \begin{minipage}{4.2cm}
404 Agglomeration of C-Si dumbbells\\[-0.1cm]
405 $\Rightarrow$ dark contrasts
409 \begin{minipage}{4cm}
411 Precipitation of 3C-SiC in Si\\[-0.1cm]
412 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
413 \& release of Si self-interstitials
417 \begin{minipage}{3.8cm}
419 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
423 \begin{minipage}{3.8cm}
425 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
429 \begin{minipage}{3.8cm}
431 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
435 \begin{pspicture}(0,0)(0,0)
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447 Molecular dynamics (MD) simulations
456 \item Microscopic description of N particle system
457 \item Analytical interaction potential
458 \item Numerical integration using Newtons equation of motion\\
459 as a propagation rule in 6N-dimensional phase space
460 \item Observables obtained by time and/or ensemble averages
462 {\bf Details of the simulation:}
464 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
465 \item Ensemble: NpT (isothermal-isobaric)
467 \item Berendsen thermostat:
468 $\tau_{\text{T}}=100\text{ fs}$
469 \item Berendsen barostat:\\
470 $\tau_{\text{P}}=100\text{ fs}$,
471 $\beta^{-1}=100\text{ GPa}$
473 \item Erhart/Albe potential: Tersoff-like bond order potential
476 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
477 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
481 \begin{picture}(0,0)(-230,-30)
482 \includegraphics[width=5cm]{tersoff_angle.eps}
490 Density functional theory (DFT) calculations
495 Basic ingredients necessary for DFT
498 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
500 \item ... uniquely determines the ground state potential
502 \item ... minimizes the systems total energy
504 \item \underline{Born-Oppenheimer}
505 - $N$ moving electrons in an external potential of static nuclei
507 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
508 +\sum_i^N V_{\text{ext}}(r_i)
509 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
511 \item \underline{Effective potential}
512 - averaged electrostatic potential \& exchange and correlation
514 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
517 \item \underline{Kohn-Sham system}
518 - Schr\"odinger equation of N non-interacting particles
520 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
525 n(r)=\sum_i^N|\Phi_i(r)|^2
527 \item \underline{Self-consistent solution}\\
528 $n(r)$ depends on $\Phi_i$, which depends on $V_{\text{eff}}$,
529 which in turn depends on $n(r)$
530 \item \underline{Variational principle}
531 - minimize total energy with respect to $n(r)$
539 Density functional theory (DFT) calculations
546 Details of applied DFT calculations in this work
549 \item \underline{Exchange correlation functional}
550 - approximations for the inhomogeneous electron gas
552 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
553 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
555 \item \underline{Plane wave basis set}
556 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
559 \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}}
561 \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
562 \item \underline{Pseudo potential}
563 - consider only the valence electrons
564 \item \underline{Code} - VASP 4.6
569 MD and structural optimization
572 \item MD integration: Gear predictor corrector algorithm
573 \item Pressure control: Parrinello-Rahman pressure control
574 \item Structural optimization: Conjugate gradient method
582 C and Si self-interstitial point defects in silicon
589 \begin{minipage}{8cm}
591 \begin{pspicture}(0,0)(7,5)
592 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
595 \item Creation of c-Si simulation volume
596 \item Periodic boundary conditions
597 \item $T=0\text{ K}$, $p=0\text{ bar}$
600 \rput(3.5,2.1){\rnode{insert}{\psframebox{
603 Insertion of interstitial C/Si atoms
606 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
609 Relaxation / structural energy minimization
612 \ncline[]{->}{init}{insert}
613 \ncline[]{->}{insert}{cool}
616 \begin{minipage}{5cm}
617 \includegraphics[width=5cm]{unit_cell_e.eps}\\
620 \begin{minipage}{9cm}
621 \begin{tabular}{l c c}
623 & size [unit cells] & \# atoms\\
625 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
626 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
630 \begin{minipage}{4cm}
631 {\color{red}$\bullet$} Tetrahedral\\
632 {\color{green}$\bullet$} Hexagonal\\
633 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
634 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
635 {\color{cyan}$\bullet$} Bond-centered\\
636 {\color{black}$\bullet$} Vacancy / Substitutional
645 \begin{minipage}{9.5cm}
648 Si self-interstitial point defects in silicon\\
651 \begin{tabular}{l c c c c c}
653 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
655 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
656 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
658 \end{tabular}\\[0.2cm]
660 \begin{minipage}{4.7cm}
661 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
663 \begin{minipage}{4.7cm}
665 {\tiny nearly T $\rightarrow$ T}\\
667 \includegraphics[width=4.7cm]{nhex_tet.ps}
670 \underline{Hexagonal} \hspace{2pt}
671 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
673 \begin{minipage}{2.7cm}
674 $E_{\text{f}}^*=4.48\text{ eV}$\\
675 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
677 \begin{minipage}{0.4cm}
682 \begin{minipage}{2.7cm}
683 $E_{\text{f}}=3.96\text{ eV}$\\
684 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
687 \begin{minipage}{2.9cm}
689 \underline{Vacancy}\\
690 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
695 \begin{minipage}{3.5cm}
698 \underline{\hkl<1 1 0> dumbbell}\\
699 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
700 \underline{Tetrahedral}\\
701 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
702 \underline{\hkl<1 0 0> dumbbell}\\
703 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
715 C interstitial point defects in silicon\\[-0.1cm]
718 \begin{tabular}{l c c c c c c}
720 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
722 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
723 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
725 \end{tabular}\\[0.1cm]
728 \begin{minipage}{2.7cm}
729 \underline{Hexagonal} \hspace{2pt}
730 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
731 $E_{\text{f}}^*=9.05\text{ eV}$\\
732 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
734 \begin{minipage}{0.4cm}
739 \begin{minipage}{2.7cm}
740 \underline{\hkl<1 0 0>}\\
741 $E_{\text{f}}=3.88\text{ eV}$\\
742 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
745 \begin{minipage}{2cm}
748 \begin{minipage}{3cm}
750 \underline{Tetrahedral}\\
751 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
756 \begin{minipage}{2.7cm}
757 \underline{Bond-centered}\\
758 $E_{\text{f}}^*=5.59\text{ eV}$\\
759 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
761 \begin{minipage}{0.4cm}
766 \begin{minipage}{2.7cm}
767 \underline{\hkl<1 1 0> dumbbell}\\
768 $E_{\text{f}}=5.18\text{ eV}$\\
769 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
772 \begin{minipage}{2cm}
775 \begin{minipage}{3cm}
777 \underline{Substitutional}\\
778 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
789 C \hkl<1 0 0> dumbbell interstitial configuration\\
793 \begin{tabular}{l c c c c c c c c}
795 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
797 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
798 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
800 \end{tabular}\\[0.2cm]
801 \begin{tabular}{l c c c c }
803 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
805 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
806 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
808 \end{tabular}\\[0.2cm]
809 \begin{tabular}{l c c c}
811 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
813 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
814 VASP & 0.109 & -0.065 & 0.174 \\
816 \end{tabular}\\[0.6cm]
819 \begin{minipage}{3.0cm}
821 \underline{Erhart/Albe}
822 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
825 \begin{minipage}{3.0cm}
828 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
832 \begin{picture}(0,0)(-185,10)
833 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
835 \begin{picture}(0,0)(-280,-150)
836 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
839 \begin{pspicture}(0,0)(0,0)
840 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
841 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
842 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
843 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
852 \begin{minipage}{8.5cm}
855 Bond-centered interstitial configuration\\[-0.1cm]
858 \begin{minipage}{3.0cm}
859 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
861 \begin{minipage}{5.2cm}
863 \item Linear Si-C-Si bond
864 \item Si: one C \& 3 Si neighbours
865 \item Spin polarized calculations
866 \item No saddle point!\\
873 \begin{minipage}[t]{6.5cm}
874 \begin{minipage}[t]{1.2cm}
876 {\tiny sp$^3$}\\[0.8cm]
877 \underline{${\color{black}\uparrow}$}
878 \underline{${\color{black}\uparrow}$}
879 \underline{${\color{black}\uparrow}$}
880 \underline{${\color{red}\uparrow}$}\\
883 \begin{minipage}[t]{1.4cm}
885 {\color{red}M}{\color{blue}O}\\[0.8cm]
886 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
887 $\sigma_{\text{ab}}$\\[0.5cm]
888 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
892 \begin{minipage}[t]{1.0cm}
896 \underline{${\color{white}\uparrow\uparrow}$}
897 \underline{${\color{white}\uparrow\uparrow}$}\\
899 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
900 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
904 \begin{minipage}[t]{1.4cm}
906 {\color{blue}M}{\color{green}O}\\[0.8cm]
907 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
908 $\sigma_{\text{ab}}$\\[0.5cm]
909 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
913 \begin{minipage}[t]{1.2cm}
916 {\tiny sp$^3$}\\[0.8cm]
917 \underline{${\color{green}\uparrow}$}
918 \underline{${\color{black}\uparrow}$}
919 \underline{${\color{black}\uparrow}$}
920 \underline{${\color{black}\uparrow}$}\\
928 \begin{minipage}{4.5cm}
929 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
931 \begin{minipage}{3.5cm}
932 {\color{gray}$\bullet$} Spin up\\
933 {\color{green}$\bullet$} Spin down\\
934 {\color{blue}$\bullet$} Resulting spin up\\
935 {\color{yellow}$\bullet$} Si atoms\\
936 {\color{red}$\bullet$} C atom
941 \begin{minipage}{4.2cm}
943 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
944 {\color{green}$\Box$} {\tiny unoccupied}\\
945 {\color{red}$\bullet$} {\tiny occupied}
954 Migration of the C \hkl<1 0 0> dumbbell interstitial
959 {\small Investigated pathways}
961 \begin{minipage}{8.5cm}
962 \begin{minipage}{8.3cm}
963 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
964 \begin{minipage}{2.4cm}
965 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
967 \begin{minipage}{0.4cm}
970 \begin{minipage}{2.4cm}
971 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
973 \begin{minipage}{0.4cm}
976 \begin{minipage}{2.4cm}
977 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
980 \begin{minipage}{8.3cm}
981 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
982 \begin{minipage}{2.4cm}
983 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
985 \begin{minipage}{0.4cm}
988 \begin{minipage}{2.4cm}
989 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
991 \begin{minipage}{0.4cm}
994 \begin{minipage}{2.4cm}
995 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
998 \begin{minipage}{8.3cm}
999 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1000 \begin{minipage}{2.4cm}
1001 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1003 \begin{minipage}{0.4cm}
1006 \begin{minipage}{2.4cm}
1007 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1009 \begin{minipage}{0.4cm}
1012 \begin{minipage}{2.4cm}
1013 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1018 \begin{minipage}{4.2cm}
1019 {\small Constrained relaxation\\
1020 technique (CRT) method}\\
1021 \includegraphics[width=4cm]{crt_orig.eps}
1023 \item Constrain diffusing atom
1024 \item Static constraints
1027 {\small Modifications}\\
1028 \includegraphics[width=4cm]{crt_mod.eps}
1030 \item Constrain all atoms
1031 \item Update individual\\
1042 Migration of the C \hkl<1 0 0> dumbbell interstitial
1048 \begin{minipage}{5.9cm}
1050 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1053 \begin{picture}(0,0)(60,0)
1054 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1056 \begin{picture}(0,0)(-5,0)
1057 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1059 \begin{picture}(0,0)(-55,0)
1060 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1062 \begin{picture}(0,0)(12.5,10)
1063 \includegraphics[width=1cm]{110_arrow.eps}
1065 \begin{picture}(0,0)(90,0)
1066 \includegraphics[height=0.9cm]{001_arrow.eps}
1072 \begin{minipage}{0.3cm}
1076 \begin{minipage}{5.9cm}
1078 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1081 \begin{picture}(0,0)(60,0)
1082 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1084 \begin{picture}(0,0)(5,0)
1085 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1087 \begin{picture}(0,0)(-55,0)
1088 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1090 \begin{picture}(0,0)(12.5,10)
1091 \includegraphics[width=1cm]{100_arrow.eps}
1093 \begin{picture}(0,0)(90,0)
1094 \includegraphics[height=0.9cm]{001_arrow.eps}
1104 \begin{minipage}{5.9cm}
1106 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1109 \begin{picture}(0,0)(60,0)
1110 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1112 \begin{picture}(0,0)(10,0)
1113 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1115 \begin{picture}(0,0)(-60,0)
1116 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1118 \begin{picture}(0,0)(12.5,10)
1119 \includegraphics[width=1cm]{100_arrow.eps}
1121 \begin{picture}(0,0)(90,0)
1122 \includegraphics[height=0.9cm]{001_arrow.eps}
1128 \begin{minipage}{0.3cm}
1131 \begin{minipage}{6.5cm}
1134 \item Energetically most favorable path
1137 \item Activation energy: $\approx$ 0.9 eV
1138 \item Experimental values: 0.73 ... 0.87 eV
1140 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1141 \item Reorientation (path 3)
1143 \item More likely composed of two consecutive steps of type 2
1144 \item Experimental values: 0.77 ... 0.88 eV
1146 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1155 Migration of the C \hkl<1 0 0> dumbbell interstitial
1160 \begin{minipage}{6.5cm}
1163 \begin{minipage}{5.9cm}
1165 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1168 \begin{pspicture}(0,0)(0,0)
1169 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1171 \begin{picture}(0,0)(60,-50)
1172 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1174 \begin{picture}(0,0)(5,-50)
1175 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1177 \begin{picture}(0,0)(-55,-50)
1178 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1180 \begin{picture}(0,0)(12.5,-40)
1181 \includegraphics[width=1cm]{110_arrow.eps}
1183 \begin{picture}(0,0)(90,-45)
1184 \includegraphics[height=0.9cm]{001_arrow.eps}
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1189 \begin{picture}(0,0)(60,-15)
1190 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1192 \begin{picture}(0,0)(35,-15)
1193 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1195 \begin{picture}(0,0)(-5,-15)
1196 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1198 \begin{picture}(0,0)(-55,-15)
1199 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1201 \begin{picture}(0,0)(12.5,-5)
1202 \includegraphics[width=1cm]{100_arrow.eps}
1204 \begin{picture}(0,0)(90,-15)
1205 \includegraphics[height=0.9cm]{010_arrow.eps}
1211 \begin{minipage}{5.9cm}
1214 \item Lowest activation energy: $\approx$ 2.2 eV
1215 \item 2.4 times higher than VASP
1216 \item Different pathway
1217 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1222 \begin{minipage}{6.5cm}
1225 \begin{minipage}{5.9cm}
1227 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1230 \begin{pspicture}(0,0)(0,0)
1231 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1233 \begin{picture}(0,0)(60,-5)
1234 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1236 \begin{picture}(0,0)(0,-5)
1237 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1239 \begin{picture}(0,0)(-55,-5)
1240 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1242 \begin{picture}(0,0)(12.5,5)
1243 \includegraphics[width=1cm]{100_arrow.eps}
1245 \begin{picture}(0,0)(90,0)
1246 \includegraphics[height=0.9cm]{001_arrow.eps}
1254 \begin{minipage}{5.9cm}
1255 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1266 Migrations involving the C \hkl<1 1 0> dumbbell interstitial
1275 \begin{minipage}{6.0cm}
1276 \includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
1278 \begin{minipage}{7cm}
1279 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1280 \hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
1281 \hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
1282 BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
1283 Composed of three single transitions\\[0.3cm]
1284 Activation energy of second transition slightly\\
1285 higher than direct transition (path 2)\\[0.3cm]
1286 $\Rightarrow$ very unlikely to happen
1287 \end{minipage}\\[0.2cm]
1291 \begin{minipage}{6.0cm}
1292 \includegraphics[width=6cm]{110_mig.ps}
1294 \begin{minipage}{7cm}
1295 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1296 \hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
1297 \hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
1298 \hkl<0 0 -1>\\[0.3cm]
1299 Composed of two single transitions\\[0.3cm]
1300 Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
1301 $\Rightarrow$ more readily constituting a probable transition
1309 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1319 E_{\text{f}}^{\text{defect combination}}-
1320 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1321 E_{\text{f}}^{\text{2nd defect}}
1327 \begin{tabular}{l c c c c c c}
1329 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1331 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1332 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1333 \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}\\
1334 \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}\\
1335 \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}\\
1336 \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}\\
1338 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1339 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1348 \begin{minipage}[t]{3.8cm}
1349 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1350 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1352 \begin{minipage}[t]{3.5cm}
1353 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1354 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1356 \begin{minipage}[t]{5.5cm}
1358 \item Restricted to VASP simulations
1359 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1360 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1361 \item Stress compensation / increase
1362 \item Most favorable: C clustering
1363 \item Unfavored: antiparallel orientations
1364 \item Indication of energetically favored\\
1369 \begin{picture}(0,0)(-295,-130)
1370 \includegraphics[width=3.5cm]{comb_pos.eps}
1378 Combinations of C-Si \hkl<1 0 0>-type interstitials
1385 Energetically most favorable combinations along \hkl<1 1 0>
1388 \begin{tabular}{l c c c c c c}
1390 & 1 & 2 & 3 & 4 & 5 & 6\\
1392 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1393 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1394 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>\\
1401 \begin{minipage}{7.0cm}
1402 \includegraphics[width=7cm,draft=false]{db_along_110_cc.ps}
1404 \begin{minipage}{6.0cm}
1406 \item Interaction proportional to reciprocal cube of C-C distance
1417 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1427 Migration of combined defects
1432 present (describe) two starting confs, i.e. vac in c-Si
1434 present migration results $\rightarrow$ SiC
1441 Silicon carbide precipitation simulations
1446 restricted to classical MD
1452 1. temperature as in exps
1454 2. exkurs: limitations of conv...
1456 3. increased temp ... high and low
1458 4. temperature limit
1467 Silicon carbide precipitation simulations
1477 Investigation of a silicon carbide precipitate in silicon