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41 \usepackage{semlayer} % Seminar overlays
42 \usepackage{slidesec} % Seminar sections and list of slides
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45 \input{seminar.bg2} % Unofficial bugs corrections
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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{}}}
109 \newrgbcolor{si-yellow}{.6 .6 0}
110 \newrgbcolor{hb}{0.75 0.77 0.89}
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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)
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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)
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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
688 \item \underline{Implantation step 1}\\[0.1cm]
689 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
690 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
691 {\color{blue}precipitates}
692 \item \underline{Implantation step 2}\\[0.1cm]
693 Little remaining dose | \unit[180]{keV} | \degc{250}\\
695 Destruction/Amorphization of precipitates at layer interface
696 \item \underline{Annealing}\\[0.1cm]
697 \unit[10]{h} at \degc{1250}\\
698 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
702 \begin{minipage}{7cm}
703 \includegraphics[width=7cm]{ibs_3c-sic.eps}
705 \begin{minipage}{5cm}
706 \begin{pspicture}(0,0)(0,0)
708 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
709 \begin{minipage}{5.3cm}
712 3C-SiC precipitation\\
713 not yet fully understood
717 \renewcommand\labelitemi{$\Rightarrow$}
718 Details of the SiC precipitation
720 \item significant technological progress\\
721 in SiC thin film formation
722 \item perspectives for processes relying\\
723 upon prevention of SiC precipitation
727 \rput(-6.8,5.4){\pnode{h0}}
728 \rput(-3.0,5.4){\pnode{h1}}
729 \ncline[linecolor=blue]{-}{h0}{h1}
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747 Model displaying the formation of ordered lamellae
756 Supposed precipitation mechanism of SiC in Si
763 \begin{minipage}{3.8cm}
764 Si \& SiC lattice structure\\[0.2cm]
765 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
769 \begin{minipage}{3.8cm}
771 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
775 \begin{minipage}{3.8cm}
777 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
781 \begin{minipage}{4cm}
783 C-Si dimers (dumbbells)\\[-0.1cm]
784 on Si interstitial sites
788 \begin{minipage}{4.2cm}
790 Agglomeration of C-Si dumbbells\\[-0.1cm]
791 $\Rightarrow$ dark contrasts
795 \begin{minipage}{4cm}
797 Precipitation of 3C-SiC in Si\\[-0.1cm]
798 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
799 \& release of Si self-interstitials
803 \begin{minipage}{3.8cm}
805 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
809 \begin{minipage}{3.8cm}
811 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
815 \begin{minipage}{3.8cm}
817 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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827 $4a_{\text{Si}}=5a_{\text{SiC}}$
829 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
830 \hkl(h k l) planes match
832 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
842 Supposed precipitation mechanism of SiC in Si
849 \begin{minipage}{3.8cm}
850 Si \& SiC lattice structure\\[0.2cm]
851 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
855 \begin{minipage}{3.8cm}
857 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
861 \begin{minipage}{3.8cm}
863 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
867 \begin{minipage}{4cm}
869 C-Si dimers (dumbbells)\\[-0.1cm]
870 on Si interstitial sites
874 \begin{minipage}{4.2cm}
876 Agglomeration of C-Si dumbbells\\[-0.1cm]
877 $\Rightarrow$ dark contrasts
881 \begin{minipage}{4cm}
883 Precipitation of 3C-SiC in Si\\[-0.1cm]
884 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
885 \& release of Si self-interstitials
889 \begin{minipage}{3.8cm}
891 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
895 \begin{minipage}{3.8cm}
897 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
901 \begin{minipage}{3.8cm}
903 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
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913 $4a_{\text{Si}}=5a_{\text{SiC}}$
915 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
916 \hkl(h k l) planes match
918 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
921 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
922 \begin{minipage}{10cm}
924 {\color{red}\bf Controversial views}
926 \item Implantations at high T (Nejim et al.)
928 \item Topotactic transformation based on \cs
929 \item \si{} as supply reacting with further C in cleared volume
931 \item Annealing behavior (Serre et al.)
933 \item Room temperature implants $\rightarrow$ highly mobile C
934 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
935 (indicate stable \cs{} configurations)
937 \item Strained silicon \& Si/SiC heterostructures
939 \item Coherent SiC precipitates (tensile strain)
940 \item Incoherent SiC (strain relaxation)
952 Molecular dynamics (MD) simulations
961 \item Microscopic description of N particle system
962 \item Analytical interaction potential
963 \item Numerical integration using Newtons equation of motion\\
964 as a propagation rule in 6N-dimensional phase space
965 \item Observables obtained by time and/or ensemble averages
967 {\bf Details of the simulation:}
969 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
970 \item Ensemble: NpT (isothermal-isobaric)
972 \item Berendsen thermostat:
973 $\tau_{\text{T}}=100\text{ fs}$
974 \item Berendsen barostat:\\
975 $\tau_{\text{P}}=100\text{ fs}$,
976 $\beta^{-1}=100\text{ GPa}$
978 \item Erhart/Albe potential: Tersoff-like bond order potential
981 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
982 \pot_{ij} = {\color{red}f_C(r_{ij})}
983 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
987 \begin{picture}(0,0)(-230,-30)
988 \includegraphics[width=5cm]{tersoff_angle.eps}
996 Density functional theory (DFT) calculations
1001 Basic ingredients necessary for DFT
1004 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
1006 \item ... uniquely determines the ground state potential
1008 \item ... minimizes the systems total energy
1010 \item \underline{Born-Oppenheimer}
1011 - $N$ moving electrons in an external potential of static nuclei
1013 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
1014 +\sum_i^N V_{\text{ext}}(r_i)
1015 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
1017 \item \underline{Effective potential}
1018 - averaged electrostatic potential \& exchange and correlation
1020 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1021 +V_{\text{XC}}[n(r)]
1023 \item \underline{Kohn-Sham system}
1024 - Schr\"odinger equation of N non-interacting particles
1026 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
1027 =\epsilon_i\Phi_i(r)
1031 n(r)=\sum_i^N|\Phi_i(r)|^2
1033 \item \underline{Self-consistent solution}\\
1034 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
1035 which in turn depends on $n(r)$
1036 \item \underline{Variational principle}
1037 - minimize total energy with respect to $n(r)$
1045 Density functional theory (DFT) calculations
1052 Details of applied DFT calculations in this work
1055 \item \underline{Exchange correlation functional}
1056 - approximations for the inhomogeneous electron gas
1058 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
1059 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
1061 \item \underline{Plane wave basis set}
1062 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
1065 \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}}
1066 \qquad ({\color{blue}300\text{ eV}})
1068 \item \underline{Brillouin zone sampling} -
1069 {\color{blue}$\Gamma$-point only} calculations
1070 \item \underline{Pseudo potential}
1071 - consider only the valence electrons
1072 \item \underline{Code} - VASP 4.6
1077 MD and structural optimization
1080 \item MD integration: Gear predictor corrector algorithm
1081 \item Pressure control: Parrinello-Rahman pressure control
1082 \item Structural optimization: Conjugate gradient method
1085 \begin{pspicture}(0,0)(0,0)
1086 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1094 C and Si self-interstitial point defects in silicon
1101 \begin{minipage}{8cm}
1103 \begin{pspicture}(0,0)(7,5)
1104 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1107 \item Creation of c-Si simulation volume
1108 \item Periodic boundary conditions
1109 \item $T=0\text{ K}$, $p=0\text{ bar}$
1112 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1115 Insertion of interstitial C/Si atoms
1118 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1121 Relaxation / structural energy minimization
1124 \ncline[]{->}{init}{insert}
1125 \ncline[]{->}{insert}{cool}
1128 \begin{minipage}{5cm}
1129 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1132 \begin{minipage}{9cm}
1133 \begin{tabular}{l c c}
1135 & size [unit cells] & \# atoms\\
1137 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1138 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1142 \begin{minipage}{4cm}
1143 {\color{red}$\bullet$} Tetrahedral\\
1144 {\color{green}$\bullet$} Hexagonal\\
1145 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1146 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1147 {\color{cyan}$\bullet$} Bond-centered\\
1148 {\color{black}$\bullet$} Vacancy / Substitutional
1157 \begin{minipage}{9.5cm}
1160 Si self-interstitial point defects in silicon\\
1163 \begin{tabular}{l c c c c c}
1165 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1167 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1168 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1170 \end{tabular}\\[0.2cm]
1172 \begin{minipage}{4.7cm}
1173 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1175 \begin{minipage}{4.7cm}
1177 {\tiny nearly T $\rightarrow$ T}\\
1179 \includegraphics[width=4.7cm]{nhex_tet.ps}
1182 \underline{Hexagonal} \hspace{2pt}
1183 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1185 \begin{minipage}{2.7cm}
1186 $E_{\text{f}}^*=4.48\text{ eV}$\\
1187 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1189 \begin{minipage}{0.4cm}
1194 \begin{minipage}{2.7cm}
1195 $E_{\text{f}}=3.96\text{ eV}$\\
1196 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1199 \begin{minipage}{2.9cm}
1201 \underline{Vacancy}\\
1202 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1207 \begin{minipage}{3.5cm}
1210 \underline{\hkl<1 1 0> dumbbell}\\
1211 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1212 \underline{Tetrahedral}\\
1213 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1214 \underline{\hkl<1 0 0> dumbbell}\\
1215 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1227 C interstitial point defects in silicon\\[-0.1cm]
1230 \begin{tabular}{l c c c c c c r}
1232 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1234 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1235 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1237 \end{tabular}\\[0.1cm]
1240 \begin{minipage}{2.7cm}
1241 \underline{Hexagonal} \hspace{2pt}
1242 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1243 $E_{\text{f}}^*=9.05\text{ eV}$\\
1244 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1246 \begin{minipage}{0.4cm}
1251 \begin{minipage}{2.7cm}
1252 \underline{\hkl<1 0 0>}\\
1253 $E_{\text{f}}=3.88\text{ eV}$\\
1254 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1257 \begin{minipage}{2cm}
1260 \begin{minipage}{3cm}
1262 \underline{Tetrahedral}\\
1263 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1268 \begin{minipage}{2.7cm}
1269 \underline{Bond-centered}\\
1270 $E_{\text{f}}^*=5.59\text{ eV}$\\
1271 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1273 \begin{minipage}{0.4cm}
1278 \begin{minipage}{2.7cm}
1279 \underline{\hkl<1 1 0> dumbbell}\\
1280 $E_{\text{f}}=5.18\text{ eV}$\\
1281 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1284 \begin{minipage}{2cm}
1287 \begin{minipage}{3cm}
1289 \underline{Substitutional}\\
1290 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1301 C \hkl<1 0 0> dumbbell interstitial configuration\\
1305 \begin{tabular}{l c c c c c c c c}
1307 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1309 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1310 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1312 \end{tabular}\\[0.2cm]
1313 \begin{tabular}{l c c c c }
1315 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1317 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1318 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1320 \end{tabular}\\[0.2cm]
1321 \begin{tabular}{l c c c}
1323 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1325 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1326 VASP & 0.109 & -0.065 & 0.174 \\
1328 \end{tabular}\\[0.6cm]
1331 \begin{minipage}{3.0cm}
1333 \underline{Erhart/Albe}
1334 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1337 \begin{minipage}{3.0cm}
1340 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1344 \begin{picture}(0,0)(-185,10)
1345 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1347 \begin{picture}(0,0)(-280,-150)
1348 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1351 \begin{pspicture}(0,0)(0,0)
1352 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1353 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1354 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1355 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1364 \begin{minipage}{8.5cm}
1367 Bond-centered interstitial configuration\\[-0.1cm]
1370 \begin{minipage}{3.0cm}
1371 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1373 \begin{minipage}{5.2cm}
1375 \item Linear Si-C-Si bond
1376 \item Si: one C \& 3 Si neighbours
1377 \item Spin polarized calculations
1378 \item No saddle point!\\
1385 \begin{minipage}[t]{6.5cm}
1386 \begin{minipage}[t]{1.2cm}
1388 {\tiny sp$^3$}\\[0.8cm]
1389 \underline{${\color{black}\uparrow}$}
1390 \underline{${\color{black}\uparrow}$}
1391 \underline{${\color{black}\uparrow}$}
1392 \underline{${\color{red}\uparrow}$}\\
1395 \begin{minipage}[t]{1.4cm}
1397 {\color{red}M}{\color{blue}O}\\[0.8cm]
1398 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1399 $\sigma_{\text{ab}}$\\[0.5cm]
1400 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1404 \begin{minipage}[t]{1.0cm}
1408 \underline{${\color{white}\uparrow\uparrow}$}
1409 \underline{${\color{white}\uparrow\uparrow}$}\\
1411 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1412 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1416 \begin{minipage}[t]{1.4cm}
1418 {\color{blue}M}{\color{green}O}\\[0.8cm]
1419 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1420 $\sigma_{\text{ab}}$\\[0.5cm]
1421 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1425 \begin{minipage}[t]{1.2cm}
1428 {\tiny sp$^3$}\\[0.8cm]
1429 \underline{${\color{green}\uparrow}$}
1430 \underline{${\color{black}\uparrow}$}
1431 \underline{${\color{black}\uparrow}$}
1432 \underline{${\color{black}\uparrow}$}\\
1440 \begin{minipage}{4.5cm}
1441 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1443 \begin{minipage}{3.5cm}
1444 {\color{gray}$\bullet$} Spin up\\
1445 {\color{green}$\bullet$} Spin down\\
1446 {\color{blue}$\bullet$} Resulting spin up\\
1447 {\color{yellow}$\bullet$} Si atoms\\
1448 {\color{red}$\bullet$} C atom
1453 \begin{minipage}{4.2cm}
1455 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1456 {\color{green}$\Box$} {\tiny unoccupied}\\
1457 {\color{red}$\bullet$} {\tiny occupied}
1466 Migration of the C \hkl<1 0 0> dumbbell interstitial
1471 {\small Investigated pathways}
1473 \begin{minipage}{8.5cm}
1474 \begin{minipage}{8.3cm}
1475 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1476 \begin{minipage}{2.4cm}
1477 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1479 \begin{minipage}{0.4cm}
1482 \begin{minipage}{2.4cm}
1483 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1485 \begin{minipage}{0.4cm}
1488 \begin{minipage}{2.4cm}
1489 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1492 \begin{minipage}{8.3cm}
1493 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1494 \begin{minipage}{2.4cm}
1495 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1497 \begin{minipage}{0.4cm}
1500 \begin{minipage}{2.4cm}
1501 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1503 \begin{minipage}{0.4cm}
1506 \begin{minipage}{2.4cm}
1507 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1510 \begin{minipage}{8.3cm}
1511 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1512 \begin{minipage}{2.4cm}
1513 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1515 \begin{minipage}{0.4cm}
1518 \begin{minipage}{2.4cm}
1519 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1521 \begin{minipage}{0.4cm}
1524 \begin{minipage}{2.4cm}
1525 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1530 \begin{minipage}{4.2cm}
1531 {\small Constrained relaxation\\
1532 technique (CRT) method}\\
1533 \includegraphics[width=4cm]{crt_orig.eps}
1535 \item Constrain diffusing atom
1536 \item Static constraints
1539 {\small Modifications}\\
1540 \includegraphics[width=4cm]{crt_mod.eps}
1542 \item Constrain all atoms
1543 \item Update individual\\
1554 Migration of the C \hkl<1 0 0> dumbbell interstitial
1560 \begin{minipage}{5.9cm}
1562 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1565 \begin{picture}(0,0)(60,0)
1566 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1568 \begin{picture}(0,0)(-5,0)
1569 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1571 \begin{picture}(0,0)(-55,0)
1572 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1574 \begin{picture}(0,0)(12.5,10)
1575 \includegraphics[width=1cm]{110_arrow.eps}
1577 \begin{picture}(0,0)(90,0)
1578 \includegraphics[height=0.9cm]{001_arrow.eps}
1584 \begin{minipage}{0.3cm}
1588 \begin{minipage}{5.9cm}
1590 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1593 \begin{picture}(0,0)(60,0)
1594 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1596 \begin{picture}(0,0)(5,0)
1597 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1599 \begin{picture}(0,0)(-55,0)
1600 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1602 \begin{picture}(0,0)(12.5,10)
1603 \includegraphics[width=1cm]{100_arrow.eps}
1605 \begin{picture}(0,0)(90,0)
1606 \includegraphics[height=0.9cm]{001_arrow.eps}
1616 \begin{minipage}{5.9cm}
1618 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1621 \begin{picture}(0,0)(60,0)
1622 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1624 \begin{picture}(0,0)(10,0)
1625 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1627 \begin{picture}(0,0)(-60,0)
1628 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1630 \begin{picture}(0,0)(12.5,10)
1631 \includegraphics[width=1cm]{100_arrow.eps}
1633 \begin{picture}(0,0)(90,0)
1634 \includegraphics[height=0.9cm]{001_arrow.eps}
1640 \begin{minipage}{0.3cm}
1643 \begin{minipage}{6.5cm}
1646 \item Energetically most favorable path
1649 \item Activation energy: $\approx$ 0.9 eV
1650 \item Experimental values: 0.73 ... 0.87 eV
1652 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1653 \item Reorientation (path 3)
1655 \item More likely composed of two consecutive steps of type 2
1656 \item Experimental values: 0.77 ... 0.88 eV
1658 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1667 Migration of the C \hkl<1 0 0> dumbbell interstitial
1674 \begin{minipage}{6.5cm}
1677 \begin{minipage}[t]{5.9cm}
1679 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1682 \begin{pspicture}(0,0)(0,0)
1683 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1685 \begin{picture}(0,0)(60,-50)
1686 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1688 \begin{picture}(0,0)(5,-50)
1689 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1691 \begin{picture}(0,0)(-55,-50)
1692 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1694 \begin{picture}(0,0)(12.5,-40)
1695 \includegraphics[width=1cm]{110_arrow.eps}
1697 \begin{picture}(0,0)(90,-45)
1698 \includegraphics[height=0.9cm]{001_arrow.eps}
1700 \begin{pspicture}(0,0)(0,0)
1701 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1703 \begin{picture}(0,0)(60,-15)
1704 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1706 \begin{picture}(0,0)(35,-15)
1707 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1709 \begin{picture}(0,0)(-5,-15)
1710 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1712 \begin{picture}(0,0)(-55,-15)
1713 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1715 \begin{picture}(0,0)(12.5,-5)
1716 \includegraphics[width=1cm]{100_arrow.eps}
1718 \begin{picture}(0,0)(90,-15)
1719 \includegraphics[height=0.9cm]{010_arrow.eps}
1725 \begin{minipage}{5.9cm}
1728 \item Lowest activation energy: $\approx$ 2.2 eV
1729 \item 2.4 times higher than VASP
1730 \item Different pathway
1735 \begin{minipage}{6.5cm}
1738 \begin{minipage}{5.9cm}
1740 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1743 %\begin{pspicture}(0,0)(0,0)
1744 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1746 %\begin{picture}(0,0)(60,-5)
1747 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1749 %\begin{picture}(0,0)(0,-5)
1750 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1752 %\begin{picture}(0,0)(-55,-5)
1753 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1755 %\begin{picture}(0,0)(12.5,5)
1756 %\includegraphics[width=1cm]{100_arrow.eps}
1758 %\begin{picture}(0,0)(90,0)
1759 %\includegraphics[height=0.9cm]{001_arrow.eps}
1767 %\begin{minipage}{5.9cm}
1768 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1772 \begin{minipage}{5.9cm}
1773 Transition involving \ci{} \hkl<1 1 0>
1775 \item Bond-centered configuration unstable\\
1776 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1777 \item Transition minima of path 2 \& 3\\
1778 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1779 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1780 \item 2.4 - 3.4 times higher than VASP
1781 \item Rotation of dumbbell orientation
1785 {\color{blue}Overestimated diffusion barrier}
1796 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1806 E_{\text{f}}^{\text{defect combination}}-
1807 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1808 E_{\text{f}}^{\text{2nd defect}}
1814 \begin{tabular}{l c c c c c c}
1816 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1818 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1819 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1820 \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}\\
1821 \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}\\
1822 \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}\\
1823 \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}\\
1825 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1826 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1835 \begin{minipage}[t]{3.8cm}
1836 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1837 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1839 \begin{minipage}[t]{3.5cm}
1840 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1841 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1843 \begin{minipage}[t]{5.5cm}
1845 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1846 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1847 \item Stress compensation / increase
1848 \item Unfavored: antiparallel orientations
1849 \item Indication of energetically favored\\
1851 \item Most favorable: C clustering
1852 \item However: High barrier ($>4\,\text{eV}$)
1853 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1858 \begin{picture}(0,0)(-295,-130)
1859 \includegraphics[width=3.5cm]{comb_pos.eps}
1867 Combinations of C-Si \hkl<1 0 0>-type interstitials
1874 Energetically most favorable combinations along \hkl<1 1 0>
1879 \begin{tabular}{l c c c c c c}
1881 & 1 & 2 & 3 & 4 & 5 & 6\\
1883 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1884 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1885 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>\\
1892 \begin{minipage}{7.0cm}
1893 \includegraphics[width=7cm]{db_along_110_cc.ps}
1895 \begin{minipage}{6.0cm}
1897 \item Interaction proportional to reciprocal cube of C-C distance
1898 \item Saturation in the immediate vicinity
1899 \renewcommand\labelitemi{$\Rightarrow$}
1900 \item Agglomeration of \ci{} expected
1901 \item Absence of C clustering
1905 Consisten with initial precipitation model
1917 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1923 %\begin{minipage}{3.2cm}
1924 %\includegraphics[width=3cm]{sub_110_combo.eps}
1926 %\begin{minipage}{7.8cm}
1927 %\begin{tabular}{l c c c c c c}
1929 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1930 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1932 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1933 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1934 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1935 %4 & \RM{4} & B & D & E & E & D \\
1936 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1943 %\begin{tabular}{l c c c c c c c c c c}
1945 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1947 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1948 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1949 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1954 \begin{minipage}{6.0cm}
1955 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1957 \begin{minipage}{7cm}
1960 \item IBS: C may displace Si\\
1961 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1963 \hkl<1 1 0>-type $\rightarrow$ favored combination
1964 \renewcommand\labelitemi{$\Rightarrow$}
1965 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1966 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1967 \item Interaction drops quickly to zero\\
1968 $\rightarrow$ low capture radius
1972 IBS process far from equilibrium\\
1973 \cs{} \& \si{} instead of thermodynamic ground state
1978 \begin{minipage}{6.5cm}
1979 \includegraphics[width=6.0cm]{162-097.ps}
1981 \item Low migration barrier
1984 \begin{minipage}{6.5cm}
1986 Ab initio MD at \degc{900}\\
1987 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1988 $t=\unit[2230]{fs}$\\
1989 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1993 Contribution of entropy to structural formation
2002 Migration in C-Si \hkl<1 0 0> and vacancy combinations
2009 \begin{minipage}[t]{3cm}
2010 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
2011 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
2013 \begin{minipage}[t]{7cm}
2016 Low activation energies\\
2017 High activation energies for reverse processes\\
2019 {\color{blue}C$_{\text{sub}}$ very stable}\\
2023 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
2025 {\color{blue}Formation of SiC by successive substitution by C}
2029 \begin{minipage}[t]{3cm}
2030 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
2031 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
2036 \begin{minipage}{5.9cm}
2037 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
2039 \begin{picture}(0,0)(70,0)
2040 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
2042 \begin{picture}(0,0)(30,0)
2043 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
2045 \begin{picture}(0,0)(-10,0)
2046 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
2048 \begin{picture}(0,0)(-48,0)
2049 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
2051 \begin{picture}(0,0)(12.5,5)
2052 \includegraphics[width=1cm]{100_arrow.eps}
2054 \begin{picture}(0,0)(97,-10)
2055 \includegraphics[height=0.9cm]{001_arrow.eps}
2061 \begin{minipage}{0.3cm}
2065 \begin{minipage}{5.9cm}
2066 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2068 \begin{picture}(0,0)(60,0)
2069 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2071 \begin{picture}(0,0)(25,0)
2072 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2074 \begin{picture}(0,0)(-20,0)
2075 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2077 \begin{picture}(0,0)(-55,0)
2078 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2080 \begin{picture}(0,0)(12.5,5)
2081 \includegraphics[width=1cm]{100_arrow.eps}
2083 \begin{picture}(0,0)(95,0)
2084 \includegraphics[height=0.9cm]{001_arrow.eps}
2096 Conclusion of defect / migration / combined defect simulations
2105 \item Accurately described by quantum-mechanical simulations
2106 \item Less accurate description by classical potential simulations
2107 \item Underestimated formation energy of \cs{} by classical approach
2108 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2113 \item C migration pathway in Si identified
2114 \item Consistent with reorientation and diffusion experiments
2117 \item Different path and ...
2118 \item overestimated barrier by classical potential calculations
2121 Concerning the precipitation mechanism
2123 \item Agglomeration of C-Si dumbbells energetically favorable
2124 (stress compensation)
2125 \item C-Si indeed favored compared to
2126 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2127 \item Possible low interaction capture radius of
2128 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2129 \item Low barrier for
2130 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2131 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2132 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2135 {\color{blue}Results suggest increased participation of \cs}
2143 Silicon carbide precipitation simulations
2149 \begin{pspicture}(0,0)(12,6.5)
2151 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2154 \item Create c-Si volume
2155 \item Periodc boundary conditions
2156 \item Set requested $T$ and $p=0\text{ bar}$
2157 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2160 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2162 Insertion of C atoms at constant T
2164 \item total simulation volume {\pnode{in1}}
2165 \item volume of minimal SiC precipitate {\pnode{in2}}
2166 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2170 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2172 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2174 \ncline[]{->}{init}{insert}
2175 \ncline[]{->}{insert}{cool}
2176 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2177 \rput(7.8,6){\footnotesize $V_1$}
2178 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2179 \rput(9.2,4.85){\tiny $V_2$}
2180 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2181 \rput(9.55,4.45){\footnotesize $V_3$}
2182 \rput(7.9,3.2){\pnode{ins1}}
2183 \rput(9.22,2.8){\pnode{ins2}}
2184 \rput(11.0,2.4){\pnode{ins3}}
2185 \ncline[]{->}{in1}{ins1}
2186 \ncline[]{->}{in2}{ins2}
2187 \ncline[]{->}{in3}{ins3}
2192 \item Restricted to classical potential simulations
2193 \item $V_2$ and $V_3$ considered due to low diffusion
2194 \item Amount of C atoms: 6000
2195 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2196 \item Simulation volume: $31\times 31\times 31$ unit cells
2205 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2210 \begin{minipage}{6.5cm}
2211 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2213 \begin{minipage}{6.5cm}
2214 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2217 \begin{minipage}{6.5cm}
2218 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2220 \begin{minipage}{6.5cm}
2222 \underline{Low C concentration ($V_1$)}\\
2223 \hkl<1 0 0> C-Si dumbbell dominated structure
2225 \item Si-C bumbs around 0.19 nm
2226 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2227 concatenated dumbbells of various orientation
2228 \item Si-Si NN distance stretched to 0.3 nm
2230 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2231 \underline{High C concentration ($V_2$, $V_3$)}\\
2232 High amount of strongly bound C-C bonds\\
2233 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2234 Only short range order observable\\
2235 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2243 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2248 \begin{minipage}{6.5cm}
2249 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2251 \begin{minipage}{6.5cm}
2252 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2255 \begin{minipage}{6.5cm}
2256 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2258 \begin{minipage}{6.5cm}
2260 \underline{Low C concentration ($V_1$)}\\
2261 \hkl<1 0 0> C-Si dumbbell dominated structure
2263 \item Si-C bumbs around 0.19 nm
2264 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2265 concatenated dumbbells of various orientation
2266 \item Si-Si NN distance stretched to 0.3 nm
2268 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2269 \underline{High C concentration ($V_2$, $V_3$)}\\
2270 High amount of strongly bound C-C bonds\\
2271 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2272 Only short range order observable\\
2273 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2276 \begin{pspicture}(0,0)(0,0)
2277 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2278 \begin{minipage}{10cm}
2280 {\color{red}\bf 3C-SiC formation fails to appear}
2282 \item Low C concentration simulations
2284 \item Formation of \ci{} indeed occurs
2285 \item Agllomeration not observed
2287 \item High C concentration simulations
2289 \item Amorphous SiC-like structure\\
2290 (not expected at prevailing temperatures)
2291 \item Rearrangement and transition into 3C-SiC structure missing
2303 Limitations of molecular dynamics and short range potentials
2310 \underline{Time scale problem of MD}\\[0.2cm]
2311 Minimize integration error\\
2312 $\Rightarrow$ discretization considerably smaller than
2313 reciprocal of fastest vibrational mode\\[0.1cm]
2314 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2315 $\Rightarrow$ suitable choice of time step:
2316 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2317 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2318 Several local minima in energy surface separated by large energy barriers\\
2319 $\Rightarrow$ transition event corresponds to a multiple
2320 of vibrational periods\\
2321 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2322 infrequent transition events\\[0.1cm]
2323 {\color{blue}Accelerated methods:}
2324 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2328 \underline{Limitations related to the short range potential}\\[0.2cm]
2329 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2330 and 2$^{\text{nd}}$ next neighbours\\
2331 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2337 Potential enhanced problem of slow phase space propagation
2342 \underline{Approach to the (twofold) problem}\\[0.2cm]
2343 Increased temperature simulations without TAD corrections\\
2344 (accelerated methods or higher time scales exclusively not sufficient)
2346 \begin{picture}(0,0)(-260,-30)
2348 \begin{minipage}{4.2cm}
2355 \item 3C-SiC also observed for higher T
2356 \item higher T inside sample
2357 \item structural evolution vs.\\
2358 equilibrium properties
2364 \begin{picture}(0,0)(-305,-155)
2366 \begin{minipage}{2.5cm}
2370 thermodynmic sampling
2381 Increased temperature simulations at low C concentration
2386 \begin{minipage}{6.5cm}
2387 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2389 \begin{minipage}{6.5cm}
2390 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2393 \begin{minipage}{6.5cm}
2394 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2396 \begin{minipage}{6.5cm}
2398 \underline{Si-C bonds:}
2400 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2401 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2403 \underline{Si-Si bonds:}
2404 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2405 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2406 \underline{C-C bonds:}
2408 \item C-C next neighbour pairs reduced (mandatory)
2409 \item Peak at 0.3 nm slightly shifted
2411 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2412 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2414 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2416 \item Range [|-$\downarrow$]:
2417 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2418 with nearby Si$_{\text{I}}$}
2423 \begin{picture}(0,0)(-330,-74)
2426 \begin{minipage}{1.6cm}
2429 stretched SiC\\[-0.1cm]
2441 Increased temperature simulations at low C concentration
2446 \begin{minipage}{6.5cm}
2447 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2449 \begin{minipage}{6.5cm}
2450 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2453 \begin{minipage}{6.5cm}
2454 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2456 \begin{minipage}{6.5cm}
2458 \underline{Si-C bonds:}
2460 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2461 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2463 \underline{Si-Si bonds:}
2464 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2465 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2466 \underline{C-C bonds:}
2468 \item C-C next neighbour pairs reduced (mandatory)
2469 \item Peak at 0.3 nm slightly shifted
2471 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2472 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2474 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2476 \item Range [|-$\downarrow$]:
2477 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2478 with nearby Si$_{\text{I}}$}
2483 %\begin{picture}(0,0)(-330,-74)
2486 %\begin{minipage}{1.6cm}
2489 %stretched SiC\\[-0.1cm]
2496 \begin{pspicture}(0,0)(0,0)
2497 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2498 \begin{minipage}{10cm}
2500 {\color{blue}\bf Stretched SiC in c-Si}
2502 \item Consistent to precipitation model involving \cs{}
2503 \item Explains annealing behavior of high/low T C implants
2505 \item Low T: highly mobiel \ci{}
2506 \item High T: stable configurations of \cs{}
2509 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2510 $\Rightarrow$ Precipitation mechanism involving \cs{}
2520 Increased temperature simulations at high C concentration
2525 \begin{minipage}{6.5cm}
2526 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2528 \begin{minipage}{6.5cm}
2529 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2537 \begin{minipage}[t]{6.0cm}
2538 0.186 nm: Si-C pairs $\uparrow$\\
2539 (as expected in 3C-SiC)\\[0.2cm]
2540 0.282 nm: Si-C-C\\[0.2cm]
2541 $\approx$0.35 nm: C-Si-Si
2544 \begin{minipage}{0.2cm}
2548 \begin{minipage}[t]{6.0cm}
2549 0.15 nm: C-C pairs $\uparrow$\\
2550 (as expected in graphite/diamond)\\[0.2cm]
2551 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2552 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2557 \item Decreasing cut-off artifact
2558 \item {\color{red}Amorphous} SiC-like phase remains
2559 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2560 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2569 High C \& small $V$ \& short $t$
2572 Slow restructuring due to strong C-C bonds
2575 High C \& low T implants
2586 Summary and Conclusions
2594 \begin{minipage}[t]{12.9cm}
2595 \underline{Pecipitation simulations}
2597 \item High C concentration $\rightarrow$ amorphous SiC like phase
2598 \item Problem of potential enhanced slow phase space propagation
2599 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2600 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2601 \item High T necessary to simulate IBS conditions (far from equilibrium)
2602 \item Precipitation by successive agglomeration of \cs (epitaxy)
2603 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2604 (stretched SiC, interface)
2612 \begin{minipage}{12.9cm}
2617 \item Point defects excellently / fairly well described
2619 \item C$_{\text{sub}}$ drastically underestimated by EA
2620 \item EA predicts correct ground state:
2621 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2622 \item Identified migration path explaining
2623 diffusion and reorientation experiments by DFT
2624 \item EA fails to describe \ci{} migration:
2625 Wrong path \& overestimated barrier
2627 \item Combinations of defects
2629 \item Agglomeration of point defects energetically favorable
2630 by compensation of stress
2631 \item Formation of C-C unlikely
2632 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2633 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2634 Low barrier (\unit[0.77]{eV}) \& low capture radius
2642 \framebox{Precipitation by successive agglomeration of \cs{}}
2660 \underline{Augsburg}
2662 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2663 \item Ralf Utermann (EDV)
2666 \underline{Helsinki}
2668 \item Prof. K. Nordlund (MD)
2673 \item Bayerische Forschungsstiftung (financial support)
2676 \underline{Paderborn}
2678 \item Prof. J. Lindner (SiC)
2679 \item Prof. G. Schmidt (DFT + financial support)
2680 \item Dr. E. Rauls (DFT + SiC)
2681 \item Dr. S. Sanna (VASP)
2688 \bf Thank you for your attention!