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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}
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160 \includegraphics[width=2.2cm]{infineon_schottky.eps}
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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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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)
345 \begin{minipage}{6.3cm}
348 Precipitation mechanism not yet fully understood!
350 \renewcommand\labelitemi{$\Rightarrow$}
352 \underline{Understanding the SiC precipitation}
354 \item significant technological progress in SiC thin film formation
355 \item perspectives for processes relying upon prevention of SiC precipitation
365 Supposed precipitation mechanism of SiC in Si
372 \begin{minipage}{3.8cm}
373 Si \& SiC lattice structure\\[0.2cm]
374 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
378 \begin{minipage}{3.8cm}
380 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
384 \begin{minipage}{3.8cm}
386 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
390 \begin{minipage}{4cm}
392 C-Si dimers (dumbbells)\\[-0.1cm]
393 on Si interstitial sites
397 \begin{minipage}{4.2cm}
399 Agglomeration of C-Si dumbbells\\[-0.1cm]
400 $\Rightarrow$ dark contrasts
404 \begin{minipage}{4cm}
406 Precipitation of 3C-SiC in Si\\[-0.1cm]
407 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
408 \& release of Si self-interstitials
412 \begin{minipage}{3.8cm}
414 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
418 \begin{minipage}{3.8cm}
420 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
424 \begin{minipage}{3.8cm}
426 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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442 Molecular dynamics (MD) simulations
451 \item Microscopic description of N particle system
452 \item Analytical interaction potential
453 \item Numerical integration using Newtons equation of motion\\
454 as a propagation rule in 6N-dimensional phase space
455 \item Observables obtained by time and/or ensemble averages
457 {\bf Details of the simulation:}
459 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
460 \item Ensemble: NpT (isothermal-isobaric)
462 \item Berendsen thermostat:
463 $\tau_{\text{T}}=100\text{ fs}$
464 \item Berendsen barostat:\\
465 $\tau_{\text{P}}=100\text{ fs}$,
466 $\beta^{-1}=100\text{ GPa}$
468 \item Erhart/Albe potential: Tersoff-like bond order potential
471 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
472 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
476 \begin{picture}(0,0)(-230,-30)
477 \includegraphics[width=5cm]{tersoff_angle.eps}
485 Density functional theory (DFT) calculations
490 Basic ingredients necessary for DFT
493 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
495 \item ... uniquely determines the ground state potential
497 \item ... minimizes the systems total energy
499 \item \underline{Born-Oppenheimer}
500 - $N$ moving electrons in an external potential of static nuclei
502 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
503 +\sum_i^N V_{\text{ext}}(r_i)
504 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
506 \item \underline{Effective potential}
507 - averaged electrostatic potential \& exchange and correlation
509 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
512 \item \underline{Kohn-Sham system}
513 - Schr\"odinger equation of N non-interacting particles
515 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
520 n(r)=\sum_i^N|\Phi_i(r)|^2
522 \item \underline{Self-consistent solution}\\
523 $n(r)$ depends on $\Phi_i$, which depends on $V_{\text{eff}}$,
524 which in turn depends on $n(r)$
525 \item \underline{Variational principle}
526 - minimize total energy with respect to $n(r)$
534 Density functional theory (DFT) calculations
541 Details of applied DFT calculations in this work
544 \item \underline{Exchange correlation functional}
545 - approximations for the inhomogeneous electron gas
547 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
548 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
550 \item \underline{Plane wave basis set}
551 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
554 \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}}
556 \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
557 \item \underline{Pseudo potential}
558 - consider only the valence electrons
559 \item \underline{Code} - VASP 4.6
564 MD and structural optimization
567 \item MD integration: Gear predictor corrector algorithm
568 \item Pressure control: Parrinello-Rahman pressure control
569 \item Structural optimization: Conjugate gradient method
577 C and Si self-interstitial point defects in silicon
584 \begin{minipage}{8cm}
586 \begin{pspicture}(0,0)(7,5)
587 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
590 \item Creation of c-Si simulation volume
591 \item Periodic boundary conditions
592 \item $T=0\text{ K}$, $p=0\text{ bar}$
595 \rput(3.5,2.1){\rnode{insert}{\psframebox{
598 Insertion of interstitial C/Si atoms
601 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
604 Relaxation / structural energy minimization
607 \ncline[]{->}{init}{insert}
608 \ncline[]{->}{insert}{cool}
611 \begin{minipage}{5cm}
612 \includegraphics[width=5cm]{unit_cell_e.eps}\\
615 \begin{minipage}{9cm}
616 \begin{tabular}{l c c}
618 & size [unit cells] & \# atoms\\
620 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
621 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
625 \begin{minipage}{4cm}
626 {\color{red}$\bullet$} Tetrahedral\\
627 {\color{green}$\bullet$} Hexagonal\\
628 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
629 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
630 {\color{cyan}$\bullet$} Bond-centered\\
631 {\color{black}$\bullet$} Vacancy / Substitutional
640 \begin{minipage}{9.5cm}
643 Si self-interstitial point defects in silicon\\
646 \begin{tabular}{l c c c c c}
648 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
650 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
651 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
653 \end{tabular}\\[0.2cm]
655 \begin{minipage}{4.7cm}
656 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
658 \begin{minipage}{4.7cm}
660 {\tiny nearly T $\rightarrow$ T}\\
662 \includegraphics[width=4.7cm]{nhex_tet.ps}
665 \underline{Hexagonal} \hspace{2pt}
666 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
668 \begin{minipage}{2.7cm}
669 $E_{\text{f}}^*=4.48\text{ eV}$\\
670 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
672 \begin{minipage}{0.4cm}
677 \begin{minipage}{2.7cm}
678 $E_{\text{f}}=3.96\text{ eV}$\\
679 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
682 \begin{minipage}{2.9cm}
684 \underline{Vacancy}\\
685 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
690 \begin{minipage}{3.5cm}
693 \underline{\hkl<1 1 0> dumbbell}\\
694 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
695 \underline{Tetrahedral}\\
696 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
697 \underline{\hkl<1 0 0> dumbbell}\\
698 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
710 C interstitial point defects in silicon\\[-0.1cm]
713 \begin{tabular}{l c c c c c c}
715 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
717 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
718 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
720 \end{tabular}\\[0.1cm]
723 \begin{minipage}{2.7cm}
724 \underline{Hexagonal} \hspace{2pt}
725 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
726 $E_{\text{f}}^*=9.05\text{ eV}$\\
727 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
729 \begin{minipage}{0.4cm}
734 \begin{minipage}{2.7cm}
735 \underline{\hkl<1 0 0>}\\
736 $E_{\text{f}}=3.88\text{ eV}$\\
737 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
740 \begin{minipage}{2cm}
743 \begin{minipage}{3cm}
745 \underline{Tetrahedral}\\
746 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
751 \begin{minipage}{2.7cm}
752 \underline{Bond-centered}\\
753 $E_{\text{f}}^*=5.59\text{ eV}$\\
754 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
756 \begin{minipage}{0.4cm}
761 \begin{minipage}{2.7cm}
762 \underline{\hkl<1 1 0> dumbbell}\\
763 $E_{\text{f}}=5.18\text{ eV}$\\
764 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
767 \begin{minipage}{2cm}
770 \begin{minipage}{3cm}
772 \underline{Substitutional}\\
773 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
784 C \hkl<1 0 0> dumbbell interstitial configuration\\
788 \begin{tabular}{l c c c c c c c c}
790 Distances & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
792 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
793 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
795 \end{tabular}\\[0.2cm]
796 \begin{tabular}{l c c c c }
798 Angles & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
800 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
801 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
803 \end{tabular}\\[0.2cm]
804 \begin{tabular}{l c c c}
806 Displacements & $a$ & $b$ & $|a|+|b|$ \\
808 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
809 VASP & 0.109 & -0.065 & 0.174 \\
811 \end{tabular}\\[0.6cm]
814 \begin{minipage}{3.0cm}
816 \underline{Erhart/Albe}
817 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
820 \begin{minipage}{3.0cm}
823 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
827 \begin{picture}(0,0)(-185,10)
828 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
830 \begin{picture}(0,0)(-280,-150)
831 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
841 Bond-centered interstitial configuration\\
850 Silicon carbide precipitation simulations
860 Investigation of a silicon carbide precipitate in silicon