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31 \graphicspath{{../img/}}
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41 \usepackage{semlayer} % Seminar overlays
42 \usepackage{slidesec} % Seminar sections and list of slides
44 \input{seminar.bug} % Official bugs corrections
45 \input{seminar.bg2} % Unofficial bugs corrections
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54 %\usepackage{mathptmx}
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82 \extraslideheight{10in}
87 % specify width and height
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95 \newcommand{\pot}{\mathcal{V}}
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100 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
103 \renewcommand{\phi}{\varphi}
106 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
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112 \newrgbcolor{hlbb}{0.825 0.88 0.968}
113 \newrgbcolor{lachs}{1.0 .93 .81}
116 \newcommand{\si}{Si$_{\text{i}}${}}
117 \newcommand{\ci}{C$_{\text{i}}${}}
118 \newcommand{\cs}{C$_{\text{sub}}${}}
119 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
120 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
121 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
122 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
124 % no vertical centering
135 A B C D E F G H G F E D C B A
150 Atomistic simulation studies\\[0.2cm]
156 \textsc{Frank Zirkelbach}
160 Application talk at the Max Planck Institute for Solid State Research
164 Stuttgart, November 2011
169 % no vertical centering
179 % Phase diagram of the C/Si system\\
184 \begin{minipage}{6.5cm}
185 \includegraphics[width=6.5cm]{si-c_phase.eps}
188 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
191 \begin{pspicture}(0,0)(0,0)
192 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
195 \begin{minipage}{6cm}
196 {\bf Phase diagram of the C/Si system}\\[0.2cm]
197 {\color{blue}Stoichiometric composition}
199 \item only chemical stable compound
200 \item wide band gap semiconductor\\
201 \underline{silicon carbide}, SiC
207 % motivation / properties / applications of silicon carbide
215 \begin{pspicture}(0,0)(13.5,5)
217 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
219 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
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222 \rput[lt](0,4.6){\color{gray}PROPERTIES}
224 \rput[lt](0.3,4){wide band gap}
225 \rput[lt](0.3,3.5){high electric breakdown field}
226 \rput[lt](0.3,3){good electron mobility}
227 \rput[lt](0.3,2.5){high electron saturation drift velocity}
228 \rput[lt](0.3,2){high thermal conductivity}
230 \rput[lt](0.3,1.5){hard and mechanically stable}
231 \rput[lt](0.3,1){chemically inert}
233 \rput[lt](0.3,0.5){radiation hardness}
235 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
237 \rput[rt](12.5,3.85){high-temperature, high power}
238 \rput[rt](12.5,3.5){and high-frequency}
239 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
241 \rput[rt](12.5,2.35){material suitable for extreme conditions}
242 \rput[rt](12.5,2){microelectromechanical systems}
243 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
245 \rput[rt](12.5,0.85){first wall reactor material, detectors}
246 \rput[rt](12.5,0.5){and electronic devices for space}
250 \begin{picture}(0,0)(5,-162)
251 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
253 \begin{picture}(0,0)(-120,-162)
254 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
256 \begin{picture}(0,0)(-270,-162)
257 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
260 \begin{picture}(0,0)(10,65)
261 \includegraphics[height=2.8cm]{sic_switch.eps}
263 %\begin{picture}(0,0)(-243,65)
264 \begin{picture}(0,0)(-110,65)
265 \includegraphics[height=2.8cm]{ise_99.eps}
267 %\begin{picture}(0,0)(-135,65)
268 \begin{picture}(0,0)(-100,65)
269 \includegraphics[height=1.2cm]{infineon_schottky.eps}
271 \begin{picture}(0,0)(-233,65)
272 \includegraphics[height=2.8cm]{solar_car.eps}
282 Polytypes of SiC\\[0.4cm]
285 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
286 \begin{minipage}{1.9cm}
287 {\tiny cubic (twist)}
289 \begin{minipage}{2.9cm}
290 {\tiny hexagonal (no twist)}
293 \begin{picture}(0,0)(-150,0)
294 \includegraphics[width=7cm]{polytypes.eps}
301 \begin{tabular}{l c c c c c c}
303 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
305 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
306 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
307 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
308 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
309 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
310 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
311 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
315 \begin{pspicture}(0,0)(0,0)
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321 \begin{pspicture}(0,0)(0,0)
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332 Fabrication of silicon carbide
341 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
347 SiC thin films by MBE \& CVD
349 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
350 \item \underline{Commercially available} semiconductor power devices based on
351 \underline{\foreignlanguage{greek}{a}-SiC}
352 \item Production of favored \underline{3C-SiC} material
353 \underline{less advanced}
354 \item Quality and size not yet sufficient
356 \begin{picture}(0,0)(-310,-20)
357 \includegraphics[width=2.0cm]{cree.eps}
362 Alternative approach:
363 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
370 \begin{minipage}{3.15cm}
372 \includegraphics[width=3cm]{imp.eps}\\
378 \begin{minipage}{3.15cm}
380 \includegraphics[width=3cm]{annealing.eps}\\
382 \unit[12]{h} annealing at \degc{1200}
387 \begin{minipage}{5.5cm}
388 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
391 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
403 Systematic investigation of C implantations into Si
409 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
425 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
428 \begin{pspicture}(0,0)(0,0)
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430 \begin{minipage}{11cm}
431 {\color{black}Diploma thesis}\\
432 \underline{Monte Carlo} simulation modeling the selforganization process\\
433 leading to periodic arrays of nanometric amorphous SiC precipitates
437 \begin{pspicture}(0,0)(0,0)
438 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=blue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
439 \begin{minipage}{11cm}
440 {\color{black}Doctoral studies}\\
441 Classical potential \underline{molecular dynamics} simulations \ldots\\
442 \underline{Density functional theory} calculations \ldots\\[0.2cm]
443 \ldots on defect formation and SiC precipitation in Si
447 \begin{pspicture}(0,0)(0,0)
448 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
450 \begin{pspicture}(0,0)(0,0)
451 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
460 Selforganization of nanometric amorphous SiC lamellae
468 \item Regularly spaced, nanometric spherical\\
469 and lamellar amorphous inclusions\\
470 at the upper a/c interface
471 \item Carbon accumulation\\
477 \begin{minipage}{12cm}
478 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
480 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si,
481 {\color{red}\underline{\degc{150}}},
482 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
486 \begin{picture}(0,0)(-182,-215)
487 \begin{minipage}{6.5cm}
489 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
491 XTEM bright-field and respective EFTEM C map
503 Model displaying the formation of ordered lamellae
509 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
515 \item Supersaturation of C in c-Si\\
516 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
518 \item High interfacial energy between 3C-SiC and c-Si\\
519 $\rightarrow$ {\bf Amorphous} precipitates
520 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
521 $\rightarrow$ {\bf Lateral strain} (black arrows)
522 \item Implantation range near surface\\
523 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
524 \item Reduction of the carbon supersaturation in c-Si\\
525 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
527 \item Remaining lateral strain\\
528 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
529 \item Absence of crystalline neighbours (structural information)\\
530 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
531 {\bf against recrystallization}
540 Implementation of the Monte Carlo code
546 \item \underline{Amorphization / Recrystallization}\\
547 Ion collision in discretized target determined by random numbers
548 distributed according to nuclear energy loss.
549 Amorphization/recrystallization probability:
551 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
554 \item {\color{green} $p_b$} normal `ballistic' amorphization
555 \item {\color{blue} $p_c$} carbon induced amorphization
556 \item {\color{red} $p_s$} stress enhanced amorphization
559 p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
562 \delta (\vec r) = \left\{
564 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
565 0 & \textrm{otherwise} \\
569 \item \underline{Carbon incorporation}\\
570 Incorporation volume determined according to implantation profile
571 \item \underline{Diffusion / Sputtering}
573 \item Transfer fraction of C atoms
574 of crystalline into neighbored amorphous volumes
575 \item Remove surface layer
583 \begin{minipage}{3.7cm}
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586 \begin{minipage}{3.7cm}
600 Evolution of the \ldots
605 \item lamellar precipitates
607 \ldots reproduced!\\[1.4cm]
611 Experiment \& simulation\\
612 in good agreement\\[1.0cm]
614 Simulation is able to model the whole depth region\\[1.2cm]
619 \begin{minipage}{0.5cm}
622 \begin{minipage}{8.0cm}
624 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
625 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
634 Structural \& compositional details
637 \begin{minipage}[t]{7.5cm}
638 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
640 \begin{minipage}[t]{5.0cm}
641 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
649 \item Fluctuation of C concentration in lamellae region
650 \item \unit[8--10]{at.\%} C saturation limit
651 within the respective conditions
652 \item Complementarily arranged and alternating sequence of layers\\
653 with a high and low amount of amorphous regions
654 \item C accumulation in the amorphous phase / Origin of stress
657 \begin{picture}(0,0)(-260,-50)
659 \begin{minipage}{3cm}
662 Precipitation process\\
679 Formation of epitaxial single crystalline 3C-SiC
686 \includegraphics[width=7cm]{ibs_3c-sic.eps}\\
689 \item \underline{Implantation step 1}\\[0.1cm]
690 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
691 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
692 {\color{blue}precipitates}
693 \item \underline{Implantation step 2}\\[0.1cm]
694 Little remaining dose | \unit[180]{keV} | \degc{250}\\
696 Destruction/Amorphization of precipitates at layer interface
697 \item \underline{Annealing}\\[0.1cm]
698 \unit[10]{h} at \degc{1250}\\
699 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
702 \begin{pspicture}(0,0)(0,0)
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704 \begin{minipage}{5.3cm}
707 3C-SiC precipitation\\
708 not yet fully understood
712 \renewcommand\labelitemi{$\Rightarrow$}
713 Details of the SiC precipitation
715 \item significant technological progress\\
716 in SiC thin film formation
717 \item perspectives for processes relying\\
718 upon prevention of SiC precipitation
738 Model displaying the formation of ordered lamellae
747 Supposed precipitation mechanism of SiC in Si
754 \begin{minipage}{3.8cm}
755 Si \& SiC lattice structure\\[0.2cm]
756 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
760 \begin{minipage}{3.8cm}
762 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
766 \begin{minipage}{3.8cm}
768 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
772 \begin{minipage}{4cm}
774 C-Si dimers (dumbbells)\\[-0.1cm]
775 on Si interstitial sites
779 \begin{minipage}{4.2cm}
781 Agglomeration of C-Si dumbbells\\[-0.1cm]
782 $\Rightarrow$ dark contrasts
786 \begin{minipage}{4cm}
788 Precipitation of 3C-SiC in Si\\[-0.1cm]
789 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
790 \& release of Si self-interstitials
794 \begin{minipage}{3.8cm}
796 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
800 \begin{minipage}{3.8cm}
802 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
806 \begin{minipage}{3.8cm}
808 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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818 $4a_{\text{Si}}=5a_{\text{SiC}}$
820 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
821 \hkl(h k l) planes match
823 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
833 Supposed precipitation mechanism of SiC in Si
840 \begin{minipage}{3.8cm}
841 Si \& SiC lattice structure\\[0.2cm]
842 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
846 \begin{minipage}{3.8cm}
848 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
852 \begin{minipage}{3.8cm}
854 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
858 \begin{minipage}{4cm}
860 C-Si dimers (dumbbells)\\[-0.1cm]
861 on Si interstitial sites
865 \begin{minipage}{4.2cm}
867 Agglomeration of C-Si dumbbells\\[-0.1cm]
868 $\Rightarrow$ dark contrasts
872 \begin{minipage}{4cm}
874 Precipitation of 3C-SiC in Si\\[-0.1cm]
875 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
876 \& release of Si self-interstitials
880 \begin{minipage}{3.8cm}
882 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
886 \begin{minipage}{3.8cm}
888 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
892 \begin{minipage}{3.8cm}
894 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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904 $4a_{\text{Si}}=5a_{\text{SiC}}$
906 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
907 \hkl(h k l) planes match
909 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
912 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
913 \begin{minipage}{10cm}
915 {\color{red}\bf Controversial views}
917 \item Implantations at high T (Nejim et al.)
919 \item Topotactic transformation based on \cs
920 \item \si{} as supply reacting with further C in cleared volume
922 \item Annealing behavior (Serre et al.)
924 \item Room temperature implants $\rightarrow$ highly mobile C
925 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
926 (indicate stable \cs{} configurations)
928 \item Strained silicon \& Si/SiC heterostructures
930 \item Coherent SiC precipitates (tensile strain)
931 \item Incoherent SiC (strain relaxation)
943 Molecular dynamics (MD) simulations
952 \item Microscopic description of N particle system
953 \item Analytical interaction potential
954 \item Numerical integration using Newtons equation of motion\\
955 as a propagation rule in 6N-dimensional phase space
956 \item Observables obtained by time and/or ensemble averages
958 {\bf Details of the simulation:}
960 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
961 \item Ensemble: NpT (isothermal-isobaric)
963 \item Berendsen thermostat:
964 $\tau_{\text{T}}=100\text{ fs}$
965 \item Berendsen barostat:\\
966 $\tau_{\text{P}}=100\text{ fs}$,
967 $\beta^{-1}=100\text{ GPa}$
969 \item Erhart/Albe potential: Tersoff-like bond order potential
972 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
973 \pot_{ij} = {\color{red}f_C(r_{ij})}
974 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
978 \begin{picture}(0,0)(-230,-30)
979 \includegraphics[width=5cm]{tersoff_angle.eps}
987 Density functional theory (DFT) calculations
992 Basic ingredients necessary for DFT
995 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
997 \item ... uniquely determines the ground state potential
999 \item ... minimizes the systems total energy
1001 \item \underline{Born-Oppenheimer}
1002 - $N$ moving electrons in an external potential of static nuclei
1004 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
1005 +\sum_i^N V_{\text{ext}}(r_i)
1006 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
1008 \item \underline{Effective potential}
1009 - averaged electrostatic potential \& exchange and correlation
1011 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1012 +V_{\text{XC}}[n(r)]
1014 \item \underline{Kohn-Sham system}
1015 - Schr\"odinger equation of N non-interacting particles
1017 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
1018 =\epsilon_i\Phi_i(r)
1022 n(r)=\sum_i^N|\Phi_i(r)|^2
1024 \item \underline{Self-consistent solution}\\
1025 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
1026 which in turn depends on $n(r)$
1027 \item \underline{Variational principle}
1028 - minimize total energy with respect to $n(r)$
1036 Density functional theory (DFT) calculations
1043 Details of applied DFT calculations in this work
1046 \item \underline{Exchange correlation functional}
1047 - approximations for the inhomogeneous electron gas
1049 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
1050 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
1052 \item \underline{Plane wave basis set}
1053 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
1056 \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}}
1057 \qquad ({\color{blue}300\text{ eV}})
1059 \item \underline{Brillouin zone sampling} -
1060 {\color{blue}$\Gamma$-point only} calculations
1061 \item \underline{Pseudo potential}
1062 - consider only the valence electrons
1063 \item \underline{Code} - VASP 4.6
1068 MD and structural optimization
1071 \item MD integration: Gear predictor corrector algorithm
1072 \item Pressure control: Parrinello-Rahman pressure control
1073 \item Structural optimization: Conjugate gradient method
1076 \begin{pspicture}(0,0)(0,0)
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1085 C and Si self-interstitial point defects in silicon
1092 \begin{minipage}{8cm}
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1095 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1098 \item Creation of c-Si simulation volume
1099 \item Periodic boundary conditions
1100 \item $T=0\text{ K}$, $p=0\text{ bar}$
1103 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1106 Insertion of interstitial C/Si atoms
1109 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1112 Relaxation / structural energy minimization
1115 \ncline[]{->}{init}{insert}
1116 \ncline[]{->}{insert}{cool}
1119 \begin{minipage}{5cm}
1120 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1123 \begin{minipage}{9cm}
1124 \begin{tabular}{l c c}
1126 & size [unit cells] & \# atoms\\
1128 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1129 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1133 \begin{minipage}{4cm}
1134 {\color{red}$\bullet$} Tetrahedral\\
1135 {\color{green}$\bullet$} Hexagonal\\
1136 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1137 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1138 {\color{cyan}$\bullet$} Bond-centered\\
1139 {\color{black}$\bullet$} Vacancy / Substitutional
1148 \begin{minipage}{9.5cm}
1151 Si self-interstitial point defects in silicon\\
1154 \begin{tabular}{l c c c c c}
1156 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1158 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1159 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1161 \end{tabular}\\[0.2cm]
1163 \begin{minipage}{4.7cm}
1164 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1166 \begin{minipage}{4.7cm}
1168 {\tiny nearly T $\rightarrow$ T}\\
1170 \includegraphics[width=4.7cm]{nhex_tet.ps}
1173 \underline{Hexagonal} \hspace{2pt}
1174 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1176 \begin{minipage}{2.7cm}
1177 $E_{\text{f}}^*=4.48\text{ eV}$\\
1178 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1180 \begin{minipage}{0.4cm}
1185 \begin{minipage}{2.7cm}
1186 $E_{\text{f}}=3.96\text{ eV}$\\
1187 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1190 \begin{minipage}{2.9cm}
1192 \underline{Vacancy}\\
1193 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1198 \begin{minipage}{3.5cm}
1201 \underline{\hkl<1 1 0> dumbbell}\\
1202 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1203 \underline{Tetrahedral}\\
1204 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1205 \underline{\hkl<1 0 0> dumbbell}\\
1206 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1218 C interstitial point defects in silicon\\[-0.1cm]
1221 \begin{tabular}{l c c c c c c r}
1223 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1225 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1226 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1228 \end{tabular}\\[0.1cm]
1231 \begin{minipage}{2.7cm}
1232 \underline{Hexagonal} \hspace{2pt}
1233 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1234 $E_{\text{f}}^*=9.05\text{ eV}$\\
1235 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1237 \begin{minipage}{0.4cm}
1242 \begin{minipage}{2.7cm}
1243 \underline{\hkl<1 0 0>}\\
1244 $E_{\text{f}}=3.88\text{ eV}$\\
1245 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1248 \begin{minipage}{2cm}
1251 \begin{minipage}{3cm}
1253 \underline{Tetrahedral}\\
1254 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1259 \begin{minipage}{2.7cm}
1260 \underline{Bond-centered}\\
1261 $E_{\text{f}}^*=5.59\text{ eV}$\\
1262 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1264 \begin{minipage}{0.4cm}
1269 \begin{minipage}{2.7cm}
1270 \underline{\hkl<1 1 0> dumbbell}\\
1271 $E_{\text{f}}=5.18\text{ eV}$\\
1272 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1275 \begin{minipage}{2cm}
1278 \begin{minipage}{3cm}
1280 \underline{Substitutional}\\
1281 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1292 C \hkl<1 0 0> dumbbell interstitial configuration\\
1296 \begin{tabular}{l c c c c c c c c}
1298 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1300 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1301 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1303 \end{tabular}\\[0.2cm]
1304 \begin{tabular}{l c c c c }
1306 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1308 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1309 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1311 \end{tabular}\\[0.2cm]
1312 \begin{tabular}{l c c c}
1314 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1316 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1317 VASP & 0.109 & -0.065 & 0.174 \\
1319 \end{tabular}\\[0.6cm]
1322 \begin{minipage}{3.0cm}
1324 \underline{Erhart/Albe}
1325 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1328 \begin{minipage}{3.0cm}
1331 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1335 \begin{picture}(0,0)(-185,10)
1336 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1338 \begin{picture}(0,0)(-280,-150)
1339 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1342 \begin{pspicture}(0,0)(0,0)
1343 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1344 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1345 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1346 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1355 \begin{minipage}{8.5cm}
1358 Bond-centered interstitial configuration\\[-0.1cm]
1361 \begin{minipage}{3.0cm}
1362 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1364 \begin{minipage}{5.2cm}
1366 \item Linear Si-C-Si bond
1367 \item Si: one C \& 3 Si neighbours
1368 \item Spin polarized calculations
1369 \item No saddle point!\\
1376 \begin{minipage}[t]{6.5cm}
1377 \begin{minipage}[t]{1.2cm}
1379 {\tiny sp$^3$}\\[0.8cm]
1380 \underline{${\color{black}\uparrow}$}
1381 \underline{${\color{black}\uparrow}$}
1382 \underline{${\color{black}\uparrow}$}
1383 \underline{${\color{red}\uparrow}$}\\
1386 \begin{minipage}[t]{1.4cm}
1388 {\color{red}M}{\color{blue}O}\\[0.8cm]
1389 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1390 $\sigma_{\text{ab}}$\\[0.5cm]
1391 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1395 \begin{minipage}[t]{1.0cm}
1399 \underline{${\color{white}\uparrow\uparrow}$}
1400 \underline{${\color{white}\uparrow\uparrow}$}\\
1402 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1403 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1407 \begin{minipage}[t]{1.4cm}
1409 {\color{blue}M}{\color{green}O}\\[0.8cm]
1410 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1411 $\sigma_{\text{ab}}$\\[0.5cm]
1412 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1416 \begin{minipage}[t]{1.2cm}
1419 {\tiny sp$^3$}\\[0.8cm]
1420 \underline{${\color{green}\uparrow}$}
1421 \underline{${\color{black}\uparrow}$}
1422 \underline{${\color{black}\uparrow}$}
1423 \underline{${\color{black}\uparrow}$}\\
1431 \begin{minipage}{4.5cm}
1432 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1434 \begin{minipage}{3.5cm}
1435 {\color{gray}$\bullet$} Spin up\\
1436 {\color{green}$\bullet$} Spin down\\
1437 {\color{blue}$\bullet$} Resulting spin up\\
1438 {\color{yellow}$\bullet$} Si atoms\\
1439 {\color{red}$\bullet$} C atom
1444 \begin{minipage}{4.2cm}
1446 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1447 {\color{green}$\Box$} {\tiny unoccupied}\\
1448 {\color{red}$\bullet$} {\tiny occupied}
1457 Migration of the C \hkl<1 0 0> dumbbell interstitial
1462 {\small Investigated pathways}
1464 \begin{minipage}{8.5cm}
1465 \begin{minipage}{8.3cm}
1466 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1467 \begin{minipage}{2.4cm}
1468 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1470 \begin{minipage}{0.4cm}
1473 \begin{minipage}{2.4cm}
1474 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1476 \begin{minipage}{0.4cm}
1479 \begin{minipage}{2.4cm}
1480 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1483 \begin{minipage}{8.3cm}
1484 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1485 \begin{minipage}{2.4cm}
1486 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1488 \begin{minipage}{0.4cm}
1491 \begin{minipage}{2.4cm}
1492 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1494 \begin{minipage}{0.4cm}
1497 \begin{minipage}{2.4cm}
1498 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1501 \begin{minipage}{8.3cm}
1502 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1503 \begin{minipage}{2.4cm}
1504 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1506 \begin{minipage}{0.4cm}
1509 \begin{minipage}{2.4cm}
1510 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1512 \begin{minipage}{0.4cm}
1515 \begin{minipage}{2.4cm}
1516 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1521 \begin{minipage}{4.2cm}
1522 {\small Constrained relaxation\\
1523 technique (CRT) method}\\
1524 \includegraphics[width=4cm]{crt_orig.eps}
1526 \item Constrain diffusing atom
1527 \item Static constraints
1530 {\small Modifications}\\
1531 \includegraphics[width=4cm]{crt_mod.eps}
1533 \item Constrain all atoms
1534 \item Update individual\\
1545 Migration of the C \hkl<1 0 0> dumbbell interstitial
1551 \begin{minipage}{5.9cm}
1553 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1556 \begin{picture}(0,0)(60,0)
1557 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1559 \begin{picture}(0,0)(-5,0)
1560 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1562 \begin{picture}(0,0)(-55,0)
1563 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1565 \begin{picture}(0,0)(12.5,10)
1566 \includegraphics[width=1cm]{110_arrow.eps}
1568 \begin{picture}(0,0)(90,0)
1569 \includegraphics[height=0.9cm]{001_arrow.eps}
1575 \begin{minipage}{0.3cm}
1579 \begin{minipage}{5.9cm}
1581 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1584 \begin{picture}(0,0)(60,0)
1585 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1587 \begin{picture}(0,0)(5,0)
1588 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1590 \begin{picture}(0,0)(-55,0)
1591 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1593 \begin{picture}(0,0)(12.5,10)
1594 \includegraphics[width=1cm]{100_arrow.eps}
1596 \begin{picture}(0,0)(90,0)
1597 \includegraphics[height=0.9cm]{001_arrow.eps}
1607 \begin{minipage}{5.9cm}
1609 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1612 \begin{picture}(0,0)(60,0)
1613 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1615 \begin{picture}(0,0)(10,0)
1616 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1618 \begin{picture}(0,0)(-60,0)
1619 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1621 \begin{picture}(0,0)(12.5,10)
1622 \includegraphics[width=1cm]{100_arrow.eps}
1624 \begin{picture}(0,0)(90,0)
1625 \includegraphics[height=0.9cm]{001_arrow.eps}
1631 \begin{minipage}{0.3cm}
1634 \begin{minipage}{6.5cm}
1637 \item Energetically most favorable path
1640 \item Activation energy: $\approx$ 0.9 eV
1641 \item Experimental values: 0.73 ... 0.87 eV
1643 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1644 \item Reorientation (path 3)
1646 \item More likely composed of two consecutive steps of type 2
1647 \item Experimental values: 0.77 ... 0.88 eV
1649 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1658 Migration of the C \hkl<1 0 0> dumbbell interstitial
1665 \begin{minipage}{6.5cm}
1668 \begin{minipage}[t]{5.9cm}
1670 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1673 \begin{pspicture}(0,0)(0,0)
1674 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1676 \begin{picture}(0,0)(60,-50)
1677 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1679 \begin{picture}(0,0)(5,-50)
1680 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1682 \begin{picture}(0,0)(-55,-50)
1683 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1685 \begin{picture}(0,0)(12.5,-40)
1686 \includegraphics[width=1cm]{110_arrow.eps}
1688 \begin{picture}(0,0)(90,-45)
1689 \includegraphics[height=0.9cm]{001_arrow.eps}
1691 \begin{pspicture}(0,0)(0,0)
1692 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1694 \begin{picture}(0,0)(60,-15)
1695 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1697 \begin{picture}(0,0)(35,-15)
1698 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1700 \begin{picture}(0,0)(-5,-15)
1701 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1703 \begin{picture}(0,0)(-55,-15)
1704 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1706 \begin{picture}(0,0)(12.5,-5)
1707 \includegraphics[width=1cm]{100_arrow.eps}
1709 \begin{picture}(0,0)(90,-15)
1710 \includegraphics[height=0.9cm]{010_arrow.eps}
1716 \begin{minipage}{5.9cm}
1719 \item Lowest activation energy: $\approx$ 2.2 eV
1720 \item 2.4 times higher than VASP
1721 \item Different pathway
1726 \begin{minipage}{6.5cm}
1729 \begin{minipage}{5.9cm}
1731 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1734 %\begin{pspicture}(0,0)(0,0)
1735 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1737 %\begin{picture}(0,0)(60,-5)
1738 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1740 %\begin{picture}(0,0)(0,-5)
1741 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1743 %\begin{picture}(0,0)(-55,-5)
1744 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1746 %\begin{picture}(0,0)(12.5,5)
1747 %\includegraphics[width=1cm]{100_arrow.eps}
1749 %\begin{picture}(0,0)(90,0)
1750 %\includegraphics[height=0.9cm]{001_arrow.eps}
1758 %\begin{minipage}{5.9cm}
1759 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1763 \begin{minipage}{5.9cm}
1764 Transition involving \ci{} \hkl<1 1 0>
1766 \item Bond-centered configuration unstable\\
1767 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1768 \item Transition minima of path 2 \& 3\\
1769 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1770 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1771 \item 2.4 - 3.4 times higher than VASP
1772 \item Rotation of dumbbell orientation
1776 {\color{blue}Overestimated diffusion barrier}
1787 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1797 E_{\text{f}}^{\text{defect combination}}-
1798 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1799 E_{\text{f}}^{\text{2nd defect}}
1805 \begin{tabular}{l c c c c c c}
1807 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1809 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1810 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1811 \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}\\
1812 \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}\\
1813 \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}\\
1814 \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}\\
1816 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1817 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1826 \begin{minipage}[t]{3.8cm}
1827 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1828 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1830 \begin{minipage}[t]{3.5cm}
1831 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1832 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1834 \begin{minipage}[t]{5.5cm}
1836 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1837 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1838 \item Stress compensation / increase
1839 \item Unfavored: antiparallel orientations
1840 \item Indication of energetically favored\\
1842 \item Most favorable: C clustering
1843 \item However: High barrier ($>4\,\text{eV}$)
1844 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1849 \begin{picture}(0,0)(-295,-130)
1850 \includegraphics[width=3.5cm]{comb_pos.eps}
1858 Combinations of C-Si \hkl<1 0 0>-type interstitials
1865 Energetically most favorable combinations along \hkl<1 1 0>
1870 \begin{tabular}{l c c c c c c}
1872 & 1 & 2 & 3 & 4 & 5 & 6\\
1874 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1875 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1876 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>\\
1883 \begin{minipage}{7.0cm}
1884 \includegraphics[width=7cm]{db_along_110_cc.ps}
1886 \begin{minipage}{6.0cm}
1888 \item Interaction proportional to reciprocal cube of C-C distance
1889 \item Saturation in the immediate vicinity
1890 \renewcommand\labelitemi{$\Rightarrow$}
1891 \item Agglomeration of \ci{} expected
1892 \item Absence of C clustering
1896 Consisten with initial precipitation model
1908 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1914 %\begin{minipage}{3.2cm}
1915 %\includegraphics[width=3cm]{sub_110_combo.eps}
1917 %\begin{minipage}{7.8cm}
1918 %\begin{tabular}{l c c c c c c}
1920 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1921 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1923 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1924 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1925 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1926 %4 & \RM{4} & B & D & E & E & D \\
1927 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1934 %\begin{tabular}{l c c c c c c c c c c}
1936 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1938 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1939 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1940 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1945 \begin{minipage}{6.0cm}
1946 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1948 \begin{minipage}{7cm}
1951 \item IBS: C may displace Si\\
1952 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1954 \hkl<1 1 0>-type $\rightarrow$ favored combination
1955 \renewcommand\labelitemi{$\Rightarrow$}
1956 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1957 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1958 \item Interaction drops quickly to zero\\
1959 $\rightarrow$ low capture radius
1963 IBS process far from equilibrium\\
1964 \cs{} \& \si{} instead of thermodynamic ground state
1969 \begin{minipage}{6.5cm}
1970 \includegraphics[width=6.0cm]{162-097.ps}
1972 \item Low migration barrier
1975 \begin{minipage}{6.5cm}
1977 Ab initio MD at \degc{900}\\
1978 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1979 $t=\unit[2230]{fs}$\\
1980 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1984 Contribution of entropy to structural formation
1993 Migration in C-Si \hkl<1 0 0> and vacancy combinations
2000 \begin{minipage}[t]{3cm}
2001 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
2002 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
2004 \begin{minipage}[t]{7cm}
2007 Low activation energies\\
2008 High activation energies for reverse processes\\
2010 {\color{blue}C$_{\text{sub}}$ very stable}\\
2014 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
2016 {\color{blue}Formation of SiC by successive substitution by C}
2020 \begin{minipage}[t]{3cm}
2021 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
2022 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
2027 \begin{minipage}{5.9cm}
2028 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
2030 \begin{picture}(0,0)(70,0)
2031 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
2033 \begin{picture}(0,0)(30,0)
2034 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
2036 \begin{picture}(0,0)(-10,0)
2037 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
2039 \begin{picture}(0,0)(-48,0)
2040 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
2042 \begin{picture}(0,0)(12.5,5)
2043 \includegraphics[width=1cm]{100_arrow.eps}
2045 \begin{picture}(0,0)(97,-10)
2046 \includegraphics[height=0.9cm]{001_arrow.eps}
2052 \begin{minipage}{0.3cm}
2056 \begin{minipage}{5.9cm}
2057 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2059 \begin{picture}(0,0)(60,0)
2060 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2062 \begin{picture}(0,0)(25,0)
2063 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2065 \begin{picture}(0,0)(-20,0)
2066 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2068 \begin{picture}(0,0)(-55,0)
2069 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2071 \begin{picture}(0,0)(12.5,5)
2072 \includegraphics[width=1cm]{100_arrow.eps}
2074 \begin{picture}(0,0)(95,0)
2075 \includegraphics[height=0.9cm]{001_arrow.eps}
2087 Conclusion of defect / migration / combined defect simulations
2096 \item Accurately described by quantum-mechanical simulations
2097 \item Less accurate description by classical potential simulations
2098 \item Underestimated formation energy of \cs{} by classical approach
2099 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2104 \item C migration pathway in Si identified
2105 \item Consistent with reorientation and diffusion experiments
2108 \item Different path and ...
2109 \item overestimated barrier by classical potential calculations
2112 Concerning the precipitation mechanism
2114 \item Agglomeration of C-Si dumbbells energetically favorable
2115 (stress compensation)
2116 \item C-Si indeed favored compared to
2117 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2118 \item Possible low interaction capture radius of
2119 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2120 \item Low barrier for
2121 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2122 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2123 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2126 {\color{blue}Results suggest increased participation of \cs}
2134 Silicon carbide precipitation simulations
2140 \begin{pspicture}(0,0)(12,6.5)
2142 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2145 \item Create c-Si volume
2146 \item Periodc boundary conditions
2147 \item Set requested $T$ and $p=0\text{ bar}$
2148 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2151 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2153 Insertion of C atoms at constant T
2155 \item total simulation volume {\pnode{in1}}
2156 \item volume of minimal SiC precipitate {\pnode{in2}}
2157 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2161 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2163 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2165 \ncline[]{->}{init}{insert}
2166 \ncline[]{->}{insert}{cool}
2167 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2168 \rput(7.8,6){\footnotesize $V_1$}
2169 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2170 \rput(9.2,4.85){\tiny $V_2$}
2171 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2172 \rput(9.55,4.45){\footnotesize $V_3$}
2173 \rput(7.9,3.2){\pnode{ins1}}
2174 \rput(9.22,2.8){\pnode{ins2}}
2175 \rput(11.0,2.4){\pnode{ins3}}
2176 \ncline[]{->}{in1}{ins1}
2177 \ncline[]{->}{in2}{ins2}
2178 \ncline[]{->}{in3}{ins3}
2183 \item Restricted to classical potential simulations
2184 \item $V_2$ and $V_3$ considered due to low diffusion
2185 \item Amount of C atoms: 6000
2186 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2187 \item Simulation volume: $31\times 31\times 31$ unit cells
2196 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2201 \begin{minipage}{6.5cm}
2202 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2204 \begin{minipage}{6.5cm}
2205 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2208 \begin{minipage}{6.5cm}
2209 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2211 \begin{minipage}{6.5cm}
2213 \underline{Low C concentration ($V_1$)}\\
2214 \hkl<1 0 0> C-Si dumbbell dominated structure
2216 \item Si-C bumbs around 0.19 nm
2217 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2218 concatenated dumbbells of various orientation
2219 \item Si-Si NN distance stretched to 0.3 nm
2221 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2222 \underline{High C concentration ($V_2$, $V_3$)}\\
2223 High amount of strongly bound C-C bonds\\
2224 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2225 Only short range order observable\\
2226 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2234 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2239 \begin{minipage}{6.5cm}
2240 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2242 \begin{minipage}{6.5cm}
2243 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2246 \begin{minipage}{6.5cm}
2247 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2249 \begin{minipage}{6.5cm}
2251 \underline{Low C concentration ($V_1$)}\\
2252 \hkl<1 0 0> C-Si dumbbell dominated structure
2254 \item Si-C bumbs around 0.19 nm
2255 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2256 concatenated dumbbells of various orientation
2257 \item Si-Si NN distance stretched to 0.3 nm
2259 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2260 \underline{High C concentration ($V_2$, $V_3$)}\\
2261 High amount of strongly bound C-C bonds\\
2262 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2263 Only short range order observable\\
2264 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2267 \begin{pspicture}(0,0)(0,0)
2268 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2269 \begin{minipage}{10cm}
2271 {\color{red}\bf 3C-SiC formation fails to appear}
2273 \item Low C concentration simulations
2275 \item Formation of \ci{} indeed occurs
2276 \item Agllomeration not observed
2278 \item High C concentration simulations
2280 \item Amorphous SiC-like structure\\
2281 (not expected at prevailing temperatures)
2282 \item Rearrangement and transition into 3C-SiC structure missing
2294 Limitations of molecular dynamics and short range potentials
2301 \underline{Time scale problem of MD}\\[0.2cm]
2302 Minimize integration error\\
2303 $\Rightarrow$ discretization considerably smaller than
2304 reciprocal of fastest vibrational mode\\[0.1cm]
2305 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2306 $\Rightarrow$ suitable choice of time step:
2307 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2308 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2309 Several local minima in energy surface separated by large energy barriers\\
2310 $\Rightarrow$ transition event corresponds to a multiple
2311 of vibrational periods\\
2312 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2313 infrequent transition events\\[0.1cm]
2314 {\color{blue}Accelerated methods:}
2315 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2319 \underline{Limitations related to the short range potential}\\[0.2cm]
2320 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2321 and 2$^{\text{nd}}$ next neighbours\\
2322 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2328 Potential enhanced problem of slow phase space propagation
2333 \underline{Approach to the (twofold) problem}\\[0.2cm]
2334 Increased temperature simulations without TAD corrections\\
2335 (accelerated methods or higher time scales exclusively not sufficient)
2337 \begin{picture}(0,0)(-260,-30)
2339 \begin{minipage}{4.2cm}
2346 \item 3C-SiC also observed for higher T
2347 \item higher T inside sample
2348 \item structural evolution vs.\\
2349 equilibrium properties
2355 \begin{picture}(0,0)(-305,-155)
2357 \begin{minipage}{2.5cm}
2361 thermodynmic sampling
2372 Increased temperature simulations at low C concentration
2377 \begin{minipage}{6.5cm}
2378 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2380 \begin{minipage}{6.5cm}
2381 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2384 \begin{minipage}{6.5cm}
2385 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2387 \begin{minipage}{6.5cm}
2389 \underline{Si-C bonds:}
2391 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2392 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2394 \underline{Si-Si bonds:}
2395 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2396 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2397 \underline{C-C bonds:}
2399 \item C-C next neighbour pairs reduced (mandatory)
2400 \item Peak at 0.3 nm slightly shifted
2402 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2403 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2405 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2407 \item Range [|-$\downarrow$]:
2408 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2409 with nearby Si$_{\text{I}}$}
2414 \begin{picture}(0,0)(-330,-74)
2417 \begin{minipage}{1.6cm}
2420 stretched SiC\\[-0.1cm]
2432 Increased temperature simulations at low C concentration
2437 \begin{minipage}{6.5cm}
2438 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2440 \begin{minipage}{6.5cm}
2441 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2444 \begin{minipage}{6.5cm}
2445 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2447 \begin{minipage}{6.5cm}
2449 \underline{Si-C bonds:}
2451 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2452 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2454 \underline{Si-Si bonds:}
2455 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2456 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2457 \underline{C-C bonds:}
2459 \item C-C next neighbour pairs reduced (mandatory)
2460 \item Peak at 0.3 nm slightly shifted
2462 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2463 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2465 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2467 \item Range [|-$\downarrow$]:
2468 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2469 with nearby Si$_{\text{I}}$}
2474 %\begin{picture}(0,0)(-330,-74)
2477 %\begin{minipage}{1.6cm}
2480 %stretched SiC\\[-0.1cm]
2487 \begin{pspicture}(0,0)(0,0)
2488 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2489 \begin{minipage}{10cm}
2491 {\color{blue}\bf Stretched SiC in c-Si}
2493 \item Consistent to precipitation model involving \cs{}
2494 \item Explains annealing behavior of high/low T C implants
2496 \item Low T: highly mobiel \ci{}
2497 \item High T: stable configurations of \cs{}
2500 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2501 $\Rightarrow$ Precipitation mechanism involving \cs{}
2511 Increased temperature simulations at high C concentration
2516 \begin{minipage}{6.5cm}
2517 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2519 \begin{minipage}{6.5cm}
2520 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2528 \begin{minipage}[t]{6.0cm}
2529 0.186 nm: Si-C pairs $\uparrow$\\
2530 (as expected in 3C-SiC)\\[0.2cm]
2531 0.282 nm: Si-C-C\\[0.2cm]
2532 $\approx$0.35 nm: C-Si-Si
2535 \begin{minipage}{0.2cm}
2539 \begin{minipage}[t]{6.0cm}
2540 0.15 nm: C-C pairs $\uparrow$\\
2541 (as expected in graphite/diamond)\\[0.2cm]
2542 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2543 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2548 \item Decreasing cut-off artifact
2549 \item {\color{red}Amorphous} SiC-like phase remains
2550 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2551 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2560 High C \& small $V$ \& short $t$
2563 Slow restructuring due to strong C-C bonds
2566 High C \& low T implants
2577 Summary and Conclusions
2585 \begin{minipage}[t]{12.9cm}
2586 \underline{Pecipitation simulations}
2588 \item High C concentration $\rightarrow$ amorphous SiC like phase
2589 \item Problem of potential enhanced slow phase space propagation
2590 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2591 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2592 \item High T necessary to simulate IBS conditions (far from equilibrium)
2593 \item Precipitation by successive agglomeration of \cs (epitaxy)
2594 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2595 (stretched SiC, interface)
2603 \begin{minipage}{12.9cm}
2608 \item Point defects excellently / fairly well described
2610 \item C$_{\text{sub}}$ drastically underestimated by EA
2611 \item EA predicts correct ground state:
2612 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2613 \item Identified migration path explaining
2614 diffusion and reorientation experiments by DFT
2615 \item EA fails to describe \ci{} migration:
2616 Wrong path \& overestimated barrier
2618 \item Combinations of defects
2620 \item Agglomeration of point defects energetically favorable
2621 by compensation of stress
2622 \item Formation of C-C unlikely
2623 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2624 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2625 Low barrier (\unit[0.77]{eV}) \& low capture radius
2633 \framebox{Precipitation by successive agglomeration of \cs{}}
2651 \underline{Augsburg}
2653 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2654 \item Ralf Utermann (EDV)
2657 \underline{Helsinki}
2659 \item Prof. K. Nordlund (MD)
2664 \item Bayerische Forschungsstiftung (financial support)
2667 \underline{Paderborn}
2669 \item Prof. J. Lindner (SiC)
2670 \item Prof. G. Schmidt (DFT + financial support)
2671 \item Dr. E. Rauls (DFT + SiC)
2672 \item Dr. S. Sanna (VASP)
2679 \bf Thank you for your attention!