2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
13 \usepackage{calc} % Simple computations with LaTeX variables
14 \usepackage{caption} % Improved captions
15 \usepackage{fancybox} % To have several backgrounds
17 \usepackage{fancyhdr} % Headers and footers definitions
18 \usepackage{fancyvrb} % Fancy verbatim environments
19 \usepackage{pstricks} % PSTricks with the standard color package
31 \graphicspath{{../img/}}
35 \usepackage[setpagesize=false]{hyperref}
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
52 %\usepackage{cmbright}
53 %\renewcommand{\familydefault}{\sfdefault}
54 %\usepackage{mathptmx}
58 \newcommand{\headdiplom}{
59 \begin{pspicture}(0,0)(0,0)
60 \rput(6.0,0.2){\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradmidpoint=1,linestyle=none]{
61 \begin{minipage}{14cm}
69 \newcommand{\headphd}{
70 \begin{pspicture}(0,0)(0,0)
71 \rput(6.0,0.2){\psframebox[fillstyle=gradient,gradbegin=blue,gradend=white,gradlines=1000,gradmidpoint=1,linestyle=none]{
72 \begin{minipage}{14cm}
82 \extraslideheight{10in}
87 % specify width and height
92 \def\slidetopmargin{-0.15cm}
94 \newcommand{\ham}{\mathcal{H}}
95 \newcommand{\pot}{\mathcal{V}}
96 \newcommand{\foo}{\mathcal{U}}
97 \newcommand{\vir}{\mathcal{W}}
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}
111 \newrgbcolor{lbb}{0.75 0.8 0.88}
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)
220 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
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)
316 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
318 \begin{pspicture}(0,0)(0,0)
319 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
321 \begin{pspicture}(0,0)(0,0)
322 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
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)
429 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
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, \degc{150},
481 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
485 \begin{picture}(0,0)(-182,-215)
486 \begin{minipage}{6.5cm}
488 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
490 XTEM bright-field and respective EFTEM C map
502 Model displaying the formation of ordered lamellae
508 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
514 \item Supersaturation of C in c-Si\\
515 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
517 \item High interfacial energy between 3C-SiC and c-Si\\
518 $\rightarrow$ {\bf Amorphous} precipitates
519 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
520 $\rightarrow$ {\bf Lateral strain} (black arrows)
521 \item Implantation range near surface\\
522 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
523 \item Reduction of the carbon supersaturation in c-Si\\
524 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
526 \item Remaining lateral strain\\
527 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
528 \item Absence of crystalline neighbours (structural information)\\
529 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
530 {\bf against recrystallization}
539 Implementation of the Monte Carlo code
545 \item \underline{Amorphization / Recrystallization}\\
546 Ion collision in discretized target determined by random numbers
547 distributed according to nuclear energy loss.
548 Amorphization/recrystallization probability:
550 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}}
553 \item {\color{green} $p_b$} normal `ballistic' amorphization
554 \item {\color{blue} $p_c$} carbon induced amorphization
555 \item {\color{red} $p_s$} stress enhanced amorphization
558 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{,}
561 \delta (\vec r) = \left\{
563 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
564 0 & \textrm{otherwise} \\
568 \item \underline{Carbon incorporation}\\
569 Incorporation volume determined according to implantation profile
570 \item \underline{Diffusion / Sputtering}
572 \item Transfer fraction of C atoms
573 of crystalline into neighbored amorphous volumes
574 \item Remove surface layer
584 \begin{minipage}{3.7cm}
594 Evolution of the \ldots
599 \item lamella precipitates
601 \ldots reproduced!\\[1.4cm]
605 Experiment \& simulation\\
606 in good agreement\\[1.0cm]
608 Simulation is able to model the whole depth region\\[1.2cm]
613 \begin{minipage}{0.5cm}
616 \begin{minipage}{8.0cm}
618 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
619 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
628 Structural \& compositional details
631 \begin{minipage}[t]{7.5cm}
632 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
634 \begin{minipage}[t]{5.0cm}
635 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
643 \item Fluctuation of C concentration in lamellae region
644 \item \unit[8--10]{at.\%} C saturation limit
645 within the respective conditions
646 \item Complementarily arranged and alternating sequence of layers\\
647 with a high and low amount of amorphous regions
648 \item C accumulation in the amorphous phase / Origin of stress
651 \begin{picture}(0,0)(-265,-30)
653 \begin{minipage}{3cm}
656 Precipitation process\\
678 Model displaying the formation of ordered lamellae
682 \begin{minipage}{6.3cm}
685 Precipitation mechanism not yet fully understood!
687 \renewcommand\labelitemi{$\Rightarrow$}
689 \underline{Understanding the SiC precipitation}
691 \item significant technological progress in SiC thin film formation
692 \item perspectives for processes relying upon prevention of SiC precipitation
703 Supposed precipitation mechanism of SiC in Si
710 \begin{minipage}{3.8cm}
711 Si \& SiC lattice structure\\[0.2cm]
712 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
716 \begin{minipage}{3.8cm}
718 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
722 \begin{minipage}{3.8cm}
724 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
728 \begin{minipage}{4cm}
730 C-Si dimers (dumbbells)\\[-0.1cm]
731 on Si interstitial sites
735 \begin{minipage}{4.2cm}
737 Agglomeration of C-Si dumbbells\\[-0.1cm]
738 $\Rightarrow$ dark contrasts
742 \begin{minipage}{4cm}
744 Precipitation of 3C-SiC in Si\\[-0.1cm]
745 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
746 \& release of Si self-interstitials
750 \begin{minipage}{3.8cm}
752 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
756 \begin{minipage}{3.8cm}
758 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
762 \begin{minipage}{3.8cm}
764 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
768 \begin{pspicture}(0,0)(0,0)
769 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
770 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
771 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
772 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
773 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
774 $4a_{\text{Si}}=5a_{\text{SiC}}$
776 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
777 \hkl(h k l) planes match
779 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
789 Supposed precipitation mechanism of SiC in Si
796 \begin{minipage}{3.8cm}
797 Si \& SiC lattice structure\\[0.2cm]
798 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
802 \begin{minipage}{3.8cm}
804 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
808 \begin{minipage}{3.8cm}
810 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
814 \begin{minipage}{4cm}
816 C-Si dimers (dumbbells)\\[-0.1cm]
817 on Si interstitial sites
821 \begin{minipage}{4.2cm}
823 Agglomeration of C-Si dumbbells\\[-0.1cm]
824 $\Rightarrow$ dark contrasts
828 \begin{minipage}{4cm}
830 Precipitation of 3C-SiC in Si\\[-0.1cm]
831 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
832 \& release of Si self-interstitials
836 \begin{minipage}{3.8cm}
838 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
842 \begin{minipage}{3.8cm}
844 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
848 \begin{minipage}{3.8cm}
850 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
854 \begin{pspicture}(0,0)(0,0)
855 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
856 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
857 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
858 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
859 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
860 $4a_{\text{Si}}=5a_{\text{SiC}}$
862 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
863 \hkl(h k l) planes match
865 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
868 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
869 \begin{minipage}{10cm}
871 {\color{red}\bf Controversial views}
873 \item Implantations at high T (Nejim et al.)
875 \item Topotactic transformation based on \cs
876 \item \si{} as supply reacting with further C in cleared volume
878 \item Annealing behavior (Serre et al.)
880 \item Room temperature implants $\rightarrow$ highly mobile C
881 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
882 (indicate stable \cs{} configurations)
884 \item Strained silicon \& Si/SiC heterostructures
886 \item Coherent SiC precipitates (tensile strain)
887 \item Incoherent SiC (strain relaxation)
899 Molecular dynamics (MD) simulations
908 \item Microscopic description of N particle system
909 \item Analytical interaction potential
910 \item Numerical integration using Newtons equation of motion\\
911 as a propagation rule in 6N-dimensional phase space
912 \item Observables obtained by time and/or ensemble averages
914 {\bf Details of the simulation:}
916 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
917 \item Ensemble: NpT (isothermal-isobaric)
919 \item Berendsen thermostat:
920 $\tau_{\text{T}}=100\text{ fs}$
921 \item Berendsen barostat:\\
922 $\tau_{\text{P}}=100\text{ fs}$,
923 $\beta^{-1}=100\text{ GPa}$
925 \item Erhart/Albe potential: Tersoff-like bond order potential
928 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
929 \pot_{ij} = {\color{red}f_C(r_{ij})}
930 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
934 \begin{picture}(0,0)(-230,-30)
935 \includegraphics[width=5cm]{tersoff_angle.eps}
943 Density functional theory (DFT) calculations
948 Basic ingredients necessary for DFT
951 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
953 \item ... uniquely determines the ground state potential
955 \item ... minimizes the systems total energy
957 \item \underline{Born-Oppenheimer}
958 - $N$ moving electrons in an external potential of static nuclei
960 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
961 +\sum_i^N V_{\text{ext}}(r_i)
962 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
964 \item \underline{Effective potential}
965 - averaged electrostatic potential \& exchange and correlation
967 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
970 \item \underline{Kohn-Sham system}
971 - Schr\"odinger equation of N non-interacting particles
973 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
978 n(r)=\sum_i^N|\Phi_i(r)|^2
980 \item \underline{Self-consistent solution}\\
981 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
982 which in turn depends on $n(r)$
983 \item \underline{Variational principle}
984 - minimize total energy with respect to $n(r)$
992 Density functional theory (DFT) calculations
999 Details of applied DFT calculations in this work
1002 \item \underline{Exchange correlation functional}
1003 - approximations for the inhomogeneous electron gas
1005 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
1006 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
1008 \item \underline{Plane wave basis set}
1009 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
1012 \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}}
1013 \qquad ({\color{blue}300\text{ eV}})
1015 \item \underline{Brillouin zone sampling} -
1016 {\color{blue}$\Gamma$-point only} calculations
1017 \item \underline{Pseudo potential}
1018 - consider only the valence electrons
1019 \item \underline{Code} - VASP 4.6
1024 MD and structural optimization
1027 \item MD integration: Gear predictor corrector algorithm
1028 \item Pressure control: Parrinello-Rahman pressure control
1029 \item Structural optimization: Conjugate gradient method
1032 \begin{pspicture}(0,0)(0,0)
1033 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1041 C and Si self-interstitial point defects in silicon
1048 \begin{minipage}{8cm}
1050 \begin{pspicture}(0,0)(7,5)
1051 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1054 \item Creation of c-Si simulation volume
1055 \item Periodic boundary conditions
1056 \item $T=0\text{ K}$, $p=0\text{ bar}$
1059 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1062 Insertion of interstitial C/Si atoms
1065 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1068 Relaxation / structural energy minimization
1071 \ncline[]{->}{init}{insert}
1072 \ncline[]{->}{insert}{cool}
1075 \begin{minipage}{5cm}
1076 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1079 \begin{minipage}{9cm}
1080 \begin{tabular}{l c c}
1082 & size [unit cells] & \# atoms\\
1084 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1085 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1089 \begin{minipage}{4cm}
1090 {\color{red}$\bullet$} Tetrahedral\\
1091 {\color{green}$\bullet$} Hexagonal\\
1092 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1093 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1094 {\color{cyan}$\bullet$} Bond-centered\\
1095 {\color{black}$\bullet$} Vacancy / Substitutional
1104 \begin{minipage}{9.5cm}
1107 Si self-interstitial point defects in silicon\\
1110 \begin{tabular}{l c c c c c}
1112 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1114 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1115 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1117 \end{tabular}\\[0.2cm]
1119 \begin{minipage}{4.7cm}
1120 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1122 \begin{minipage}{4.7cm}
1124 {\tiny nearly T $\rightarrow$ T}\\
1126 \includegraphics[width=4.7cm]{nhex_tet.ps}
1129 \underline{Hexagonal} \hspace{2pt}
1130 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1132 \begin{minipage}{2.7cm}
1133 $E_{\text{f}}^*=4.48\text{ eV}$\\
1134 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1136 \begin{minipage}{0.4cm}
1141 \begin{minipage}{2.7cm}
1142 $E_{\text{f}}=3.96\text{ eV}$\\
1143 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1146 \begin{minipage}{2.9cm}
1148 \underline{Vacancy}\\
1149 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1154 \begin{minipage}{3.5cm}
1157 \underline{\hkl<1 1 0> dumbbell}\\
1158 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1159 \underline{Tetrahedral}\\
1160 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1161 \underline{\hkl<1 0 0> dumbbell}\\
1162 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1174 C interstitial point defects in silicon\\[-0.1cm]
1177 \begin{tabular}{l c c c c c c r}
1179 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1181 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1182 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1184 \end{tabular}\\[0.1cm]
1187 \begin{minipage}{2.7cm}
1188 \underline{Hexagonal} \hspace{2pt}
1189 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1190 $E_{\text{f}}^*=9.05\text{ eV}$\\
1191 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1193 \begin{minipage}{0.4cm}
1198 \begin{minipage}{2.7cm}
1199 \underline{\hkl<1 0 0>}\\
1200 $E_{\text{f}}=3.88\text{ eV}$\\
1201 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1204 \begin{minipage}{2cm}
1207 \begin{minipage}{3cm}
1209 \underline{Tetrahedral}\\
1210 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1215 \begin{minipage}{2.7cm}
1216 \underline{Bond-centered}\\
1217 $E_{\text{f}}^*=5.59\text{ eV}$\\
1218 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1220 \begin{minipage}{0.4cm}
1225 \begin{minipage}{2.7cm}
1226 \underline{\hkl<1 1 0> dumbbell}\\
1227 $E_{\text{f}}=5.18\text{ eV}$\\
1228 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1231 \begin{minipage}{2cm}
1234 \begin{minipage}{3cm}
1236 \underline{Substitutional}\\
1237 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1248 C \hkl<1 0 0> dumbbell interstitial configuration\\
1252 \begin{tabular}{l c c c c c c c c}
1254 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1256 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1257 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1259 \end{tabular}\\[0.2cm]
1260 \begin{tabular}{l c c c c }
1262 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1264 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1265 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1267 \end{tabular}\\[0.2cm]
1268 \begin{tabular}{l c c c}
1270 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1272 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1273 VASP & 0.109 & -0.065 & 0.174 \\
1275 \end{tabular}\\[0.6cm]
1278 \begin{minipage}{3.0cm}
1280 \underline{Erhart/Albe}
1281 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1284 \begin{minipage}{3.0cm}
1287 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1291 \begin{picture}(0,0)(-185,10)
1292 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1294 \begin{picture}(0,0)(-280,-150)
1295 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1298 \begin{pspicture}(0,0)(0,0)
1299 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1300 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1301 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1302 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1311 \begin{minipage}{8.5cm}
1314 Bond-centered interstitial configuration\\[-0.1cm]
1317 \begin{minipage}{3.0cm}
1318 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1320 \begin{minipage}{5.2cm}
1322 \item Linear Si-C-Si bond
1323 \item Si: one C \& 3 Si neighbours
1324 \item Spin polarized calculations
1325 \item No saddle point!\\
1332 \begin{minipage}[t]{6.5cm}
1333 \begin{minipage}[t]{1.2cm}
1335 {\tiny sp$^3$}\\[0.8cm]
1336 \underline{${\color{black}\uparrow}$}
1337 \underline{${\color{black}\uparrow}$}
1338 \underline{${\color{black}\uparrow}$}
1339 \underline{${\color{red}\uparrow}$}\\
1342 \begin{minipage}[t]{1.4cm}
1344 {\color{red}M}{\color{blue}O}\\[0.8cm]
1345 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1346 $\sigma_{\text{ab}}$\\[0.5cm]
1347 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1351 \begin{minipage}[t]{1.0cm}
1355 \underline{${\color{white}\uparrow\uparrow}$}
1356 \underline{${\color{white}\uparrow\uparrow}$}\\
1358 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1359 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1363 \begin{minipage}[t]{1.4cm}
1365 {\color{blue}M}{\color{green}O}\\[0.8cm]
1366 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1367 $\sigma_{\text{ab}}$\\[0.5cm]
1368 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1372 \begin{minipage}[t]{1.2cm}
1375 {\tiny sp$^3$}\\[0.8cm]
1376 \underline{${\color{green}\uparrow}$}
1377 \underline{${\color{black}\uparrow}$}
1378 \underline{${\color{black}\uparrow}$}
1379 \underline{${\color{black}\uparrow}$}\\
1387 \begin{minipage}{4.5cm}
1388 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1390 \begin{minipage}{3.5cm}
1391 {\color{gray}$\bullet$} Spin up\\
1392 {\color{green}$\bullet$} Spin down\\
1393 {\color{blue}$\bullet$} Resulting spin up\\
1394 {\color{yellow}$\bullet$} Si atoms\\
1395 {\color{red}$\bullet$} C atom
1400 \begin{minipage}{4.2cm}
1402 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1403 {\color{green}$\Box$} {\tiny unoccupied}\\
1404 {\color{red}$\bullet$} {\tiny occupied}
1413 Migration of the C \hkl<1 0 0> dumbbell interstitial
1418 {\small Investigated pathways}
1420 \begin{minipage}{8.5cm}
1421 \begin{minipage}{8.3cm}
1422 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1423 \begin{minipage}{2.4cm}
1424 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1426 \begin{minipage}{0.4cm}
1429 \begin{minipage}{2.4cm}
1430 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1432 \begin{minipage}{0.4cm}
1435 \begin{minipage}{2.4cm}
1436 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1439 \begin{minipage}{8.3cm}
1440 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1441 \begin{minipage}{2.4cm}
1442 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1444 \begin{minipage}{0.4cm}
1447 \begin{minipage}{2.4cm}
1448 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1450 \begin{minipage}{0.4cm}
1453 \begin{minipage}{2.4cm}
1454 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1457 \begin{minipage}{8.3cm}
1458 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1459 \begin{minipage}{2.4cm}
1460 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1462 \begin{minipage}{0.4cm}
1465 \begin{minipage}{2.4cm}
1466 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1468 \begin{minipage}{0.4cm}
1471 \begin{minipage}{2.4cm}
1472 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1477 \begin{minipage}{4.2cm}
1478 {\small Constrained relaxation\\
1479 technique (CRT) method}\\
1480 \includegraphics[width=4cm]{crt_orig.eps}
1482 \item Constrain diffusing atom
1483 \item Static constraints
1486 {\small Modifications}\\
1487 \includegraphics[width=4cm]{crt_mod.eps}
1489 \item Constrain all atoms
1490 \item Update individual\\
1501 Migration of the C \hkl<1 0 0> dumbbell interstitial
1507 \begin{minipage}{5.9cm}
1509 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1512 \begin{picture}(0,0)(60,0)
1513 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1515 \begin{picture}(0,0)(-5,0)
1516 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1518 \begin{picture}(0,0)(-55,0)
1519 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1521 \begin{picture}(0,0)(12.5,10)
1522 \includegraphics[width=1cm]{110_arrow.eps}
1524 \begin{picture}(0,0)(90,0)
1525 \includegraphics[height=0.9cm]{001_arrow.eps}
1531 \begin{minipage}{0.3cm}
1535 \begin{minipage}{5.9cm}
1537 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1540 \begin{picture}(0,0)(60,0)
1541 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1543 \begin{picture}(0,0)(5,0)
1544 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1546 \begin{picture}(0,0)(-55,0)
1547 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1549 \begin{picture}(0,0)(12.5,10)
1550 \includegraphics[width=1cm]{100_arrow.eps}
1552 \begin{picture}(0,0)(90,0)
1553 \includegraphics[height=0.9cm]{001_arrow.eps}
1563 \begin{minipage}{5.9cm}
1565 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1568 \begin{picture}(0,0)(60,0)
1569 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1571 \begin{picture}(0,0)(10,0)
1572 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1574 \begin{picture}(0,0)(-60,0)
1575 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1577 \begin{picture}(0,0)(12.5,10)
1578 \includegraphics[width=1cm]{100_arrow.eps}
1580 \begin{picture}(0,0)(90,0)
1581 \includegraphics[height=0.9cm]{001_arrow.eps}
1587 \begin{minipage}{0.3cm}
1590 \begin{minipage}{6.5cm}
1593 \item Energetically most favorable path
1596 \item Activation energy: $\approx$ 0.9 eV
1597 \item Experimental values: 0.73 ... 0.87 eV
1599 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1600 \item Reorientation (path 3)
1602 \item More likely composed of two consecutive steps of type 2
1603 \item Experimental values: 0.77 ... 0.88 eV
1605 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1614 Migration of the C \hkl<1 0 0> dumbbell interstitial
1621 \begin{minipage}{6.5cm}
1624 \begin{minipage}[t]{5.9cm}
1626 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1629 \begin{pspicture}(0,0)(0,0)
1630 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1632 \begin{picture}(0,0)(60,-50)
1633 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1635 \begin{picture}(0,0)(5,-50)
1636 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1638 \begin{picture}(0,0)(-55,-50)
1639 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1641 \begin{picture}(0,0)(12.5,-40)
1642 \includegraphics[width=1cm]{110_arrow.eps}
1644 \begin{picture}(0,0)(90,-45)
1645 \includegraphics[height=0.9cm]{001_arrow.eps}
1647 \begin{pspicture}(0,0)(0,0)
1648 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1650 \begin{picture}(0,0)(60,-15)
1651 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1653 \begin{picture}(0,0)(35,-15)
1654 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1656 \begin{picture}(0,0)(-5,-15)
1657 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1659 \begin{picture}(0,0)(-55,-15)
1660 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1662 \begin{picture}(0,0)(12.5,-5)
1663 \includegraphics[width=1cm]{100_arrow.eps}
1665 \begin{picture}(0,0)(90,-15)
1666 \includegraphics[height=0.9cm]{010_arrow.eps}
1672 \begin{minipage}{5.9cm}
1675 \item Lowest activation energy: $\approx$ 2.2 eV
1676 \item 2.4 times higher than VASP
1677 \item Different pathway
1682 \begin{minipage}{6.5cm}
1685 \begin{minipage}{5.9cm}
1687 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1690 %\begin{pspicture}(0,0)(0,0)
1691 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1693 %\begin{picture}(0,0)(60,-5)
1694 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1696 %\begin{picture}(0,0)(0,-5)
1697 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1699 %\begin{picture}(0,0)(-55,-5)
1700 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1702 %\begin{picture}(0,0)(12.5,5)
1703 %\includegraphics[width=1cm]{100_arrow.eps}
1705 %\begin{picture}(0,0)(90,0)
1706 %\includegraphics[height=0.9cm]{001_arrow.eps}
1714 %\begin{minipage}{5.9cm}
1715 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1719 \begin{minipage}{5.9cm}
1720 Transition involving \ci{} \hkl<1 1 0>
1722 \item Bond-centered configuration unstable\\
1723 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1724 \item Transition minima of path 2 \& 3\\
1725 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1726 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1727 \item 2.4 - 3.4 times higher than VASP
1728 \item Rotation of dumbbell orientation
1732 {\color{blue}Overestimated diffusion barrier}
1743 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1753 E_{\text{f}}^{\text{defect combination}}-
1754 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1755 E_{\text{f}}^{\text{2nd defect}}
1761 \begin{tabular}{l c c c c c c}
1763 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1765 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1766 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1767 \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}\\
1768 \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}\\
1769 \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}\\
1770 \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}\\
1772 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1773 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1782 \begin{minipage}[t]{3.8cm}
1783 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1784 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1786 \begin{minipage}[t]{3.5cm}
1787 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1788 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1790 \begin{minipage}[t]{5.5cm}
1792 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1793 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1794 \item Stress compensation / increase
1795 \item Unfavored: antiparallel orientations
1796 \item Indication of energetically favored\\
1798 \item Most favorable: C clustering
1799 \item However: High barrier ($>4\,\text{eV}$)
1800 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1805 \begin{picture}(0,0)(-295,-130)
1806 \includegraphics[width=3.5cm]{comb_pos.eps}
1814 Combinations of C-Si \hkl<1 0 0>-type interstitials
1821 Energetically most favorable combinations along \hkl<1 1 0>
1826 \begin{tabular}{l c c c c c c}
1828 & 1 & 2 & 3 & 4 & 5 & 6\\
1830 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1831 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1832 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>\\
1839 \begin{minipage}{7.0cm}
1840 \includegraphics[width=7cm]{db_along_110_cc.ps}
1842 \begin{minipage}{6.0cm}
1844 \item Interaction proportional to reciprocal cube of C-C distance
1845 \item Saturation in the immediate vicinity
1846 \renewcommand\labelitemi{$\Rightarrow$}
1847 \item Agglomeration of \ci{} expected
1848 \item Absence of C clustering
1852 Consisten with initial precipitation model
1864 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1870 %\begin{minipage}{3.2cm}
1871 %\includegraphics[width=3cm]{sub_110_combo.eps}
1873 %\begin{minipage}{7.8cm}
1874 %\begin{tabular}{l c c c c c c}
1876 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1877 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1879 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1880 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1881 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1882 %4 & \RM{4} & B & D & E & E & D \\
1883 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1890 %\begin{tabular}{l c c c c c c c c c c}
1892 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1894 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1895 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1896 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1901 \begin{minipage}{6.0cm}
1902 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1904 \begin{minipage}{7cm}
1907 \item IBS: C may displace Si\\
1908 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1910 \hkl<1 1 0>-type $\rightarrow$ favored combination
1911 \renewcommand\labelitemi{$\Rightarrow$}
1912 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1913 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1914 \item Interaction drops quickly to zero\\
1915 $\rightarrow$ low capture radius
1919 IBS process far from equilibrium\\
1920 \cs{} \& \si{} instead of thermodynamic ground state
1925 \begin{minipage}{6.5cm}
1926 \includegraphics[width=6.0cm]{162-097.ps}
1928 \item Low migration barrier
1931 \begin{minipage}{6.5cm}
1933 Ab initio MD at \degc{900}\\
1934 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1935 $t=\unit[2230]{fs}$\\
1936 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1940 Contribution of entropy to structural formation
1949 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1956 \begin{minipage}[t]{3cm}
1957 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1958 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1960 \begin{minipage}[t]{7cm}
1963 Low activation energies\\
1964 High activation energies for reverse processes\\
1966 {\color{blue}C$_{\text{sub}}$ very stable}\\
1970 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1972 {\color{blue}Formation of SiC by successive substitution by C}
1976 \begin{minipage}[t]{3cm}
1977 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1978 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1983 \begin{minipage}{5.9cm}
1984 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1986 \begin{picture}(0,0)(70,0)
1987 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1989 \begin{picture}(0,0)(30,0)
1990 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1992 \begin{picture}(0,0)(-10,0)
1993 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1995 \begin{picture}(0,0)(-48,0)
1996 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1998 \begin{picture}(0,0)(12.5,5)
1999 \includegraphics[width=1cm]{100_arrow.eps}
2001 \begin{picture}(0,0)(97,-10)
2002 \includegraphics[height=0.9cm]{001_arrow.eps}
2008 \begin{minipage}{0.3cm}
2012 \begin{minipage}{5.9cm}
2013 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2015 \begin{picture}(0,0)(60,0)
2016 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2018 \begin{picture}(0,0)(25,0)
2019 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2021 \begin{picture}(0,0)(-20,0)
2022 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2024 \begin{picture}(0,0)(-55,0)
2025 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2027 \begin{picture}(0,0)(12.5,5)
2028 \includegraphics[width=1cm]{100_arrow.eps}
2030 \begin{picture}(0,0)(95,0)
2031 \includegraphics[height=0.9cm]{001_arrow.eps}
2043 Conclusion of defect / migration / combined defect simulations
2052 \item Accurately described by quantum-mechanical simulations
2053 \item Less accurate description by classical potential simulations
2054 \item Underestimated formation energy of \cs{} by classical approach
2055 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2060 \item C migration pathway in Si identified
2061 \item Consistent with reorientation and diffusion experiments
2064 \item Different path and ...
2065 \item overestimated barrier by classical potential calculations
2068 Concerning the precipitation mechanism
2070 \item Agglomeration of C-Si dumbbells energetically favorable
2071 (stress compensation)
2072 \item C-Si indeed favored compared to
2073 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2074 \item Possible low interaction capture radius of
2075 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2076 \item Low barrier for
2077 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2078 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2079 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2082 {\color{blue}Results suggest increased participation of \cs}
2090 Silicon carbide precipitation simulations
2096 \begin{pspicture}(0,0)(12,6.5)
2098 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2101 \item Create c-Si volume
2102 \item Periodc boundary conditions
2103 \item Set requested $T$ and $p=0\text{ bar}$
2104 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2107 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2109 Insertion of C atoms at constant T
2111 \item total simulation volume {\pnode{in1}}
2112 \item volume of minimal SiC precipitate {\pnode{in2}}
2113 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2117 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2119 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2121 \ncline[]{->}{init}{insert}
2122 \ncline[]{->}{insert}{cool}
2123 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2124 \rput(7.8,6){\footnotesize $V_1$}
2125 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2126 \rput(9.2,4.85){\tiny $V_2$}
2127 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2128 \rput(9.55,4.45){\footnotesize $V_3$}
2129 \rput(7.9,3.2){\pnode{ins1}}
2130 \rput(9.22,2.8){\pnode{ins2}}
2131 \rput(11.0,2.4){\pnode{ins3}}
2132 \ncline[]{->}{in1}{ins1}
2133 \ncline[]{->}{in2}{ins2}
2134 \ncline[]{->}{in3}{ins3}
2139 \item Restricted to classical potential simulations
2140 \item $V_2$ and $V_3$ considered due to low diffusion
2141 \item Amount of C atoms: 6000
2142 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2143 \item Simulation volume: $31\times 31\times 31$ unit cells
2152 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2157 \begin{minipage}{6.5cm}
2158 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2160 \begin{minipage}{6.5cm}
2161 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2164 \begin{minipage}{6.5cm}
2165 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2167 \begin{minipage}{6.5cm}
2169 \underline{Low C concentration ($V_1$)}\\
2170 \hkl<1 0 0> C-Si dumbbell dominated structure
2172 \item Si-C bumbs around 0.19 nm
2173 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2174 concatenated dumbbells of various orientation
2175 \item Si-Si NN distance stretched to 0.3 nm
2177 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2178 \underline{High C concentration ($V_2$, $V_3$)}\\
2179 High amount of strongly bound C-C bonds\\
2180 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2181 Only short range order observable\\
2182 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2190 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2195 \begin{minipage}{6.5cm}
2196 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2198 \begin{minipage}{6.5cm}
2199 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2202 \begin{minipage}{6.5cm}
2203 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2205 \begin{minipage}{6.5cm}
2207 \underline{Low C concentration ($V_1$)}\\
2208 \hkl<1 0 0> C-Si dumbbell dominated structure
2210 \item Si-C bumbs around 0.19 nm
2211 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2212 concatenated dumbbells of various orientation
2213 \item Si-Si NN distance stretched to 0.3 nm
2215 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2216 \underline{High C concentration ($V_2$, $V_3$)}\\
2217 High amount of strongly bound C-C bonds\\
2218 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2219 Only short range order observable\\
2220 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2223 \begin{pspicture}(0,0)(0,0)
2224 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2225 \begin{minipage}{10cm}
2227 {\color{red}\bf 3C-SiC formation fails to appear}
2229 \item Low C concentration simulations
2231 \item Formation of \ci{} indeed occurs
2232 \item Agllomeration not observed
2234 \item High C concentration simulations
2236 \item Amorphous SiC-like structure\\
2237 (not expected at prevailing temperatures)
2238 \item Rearrangement and transition into 3C-SiC structure missing
2250 Limitations of molecular dynamics and short range potentials
2257 \underline{Time scale problem of MD}\\[0.2cm]
2258 Minimize integration error\\
2259 $\Rightarrow$ discretization considerably smaller than
2260 reciprocal of fastest vibrational mode\\[0.1cm]
2261 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2262 $\Rightarrow$ suitable choice of time step:
2263 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2264 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2265 Several local minima in energy surface separated by large energy barriers\\
2266 $\Rightarrow$ transition event corresponds to a multiple
2267 of vibrational periods\\
2268 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2269 infrequent transition events\\[0.1cm]
2270 {\color{blue}Accelerated methods:}
2271 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2275 \underline{Limitations related to the short range potential}\\[0.2cm]
2276 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2277 and 2$^{\text{nd}}$ next neighbours\\
2278 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2284 Potential enhanced problem of slow phase space propagation
2289 \underline{Approach to the (twofold) problem}\\[0.2cm]
2290 Increased temperature simulations without TAD corrections\\
2291 (accelerated methods or higher time scales exclusively not sufficient)
2293 \begin{picture}(0,0)(-260,-30)
2295 \begin{minipage}{4.2cm}
2302 \item 3C-SiC also observed for higher T
2303 \item higher T inside sample
2304 \item structural evolution vs.\\
2305 equilibrium properties
2311 \begin{picture}(0,0)(-305,-155)
2313 \begin{minipage}{2.5cm}
2317 thermodynmic sampling
2328 Increased temperature simulations at low C concentration
2333 \begin{minipage}{6.5cm}
2334 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2336 \begin{minipage}{6.5cm}
2337 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2340 \begin{minipage}{6.5cm}
2341 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2343 \begin{minipage}{6.5cm}
2345 \underline{Si-C bonds:}
2347 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2348 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2350 \underline{Si-Si bonds:}
2351 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2352 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2353 \underline{C-C bonds:}
2355 \item C-C next neighbour pairs reduced (mandatory)
2356 \item Peak at 0.3 nm slightly shifted
2358 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2359 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2361 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2363 \item Range [|-$\downarrow$]:
2364 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2365 with nearby Si$_{\text{I}}$}
2370 \begin{picture}(0,0)(-330,-74)
2373 \begin{minipage}{1.6cm}
2376 stretched SiC\\[-0.1cm]
2388 Increased temperature simulations at low C concentration
2393 \begin{minipage}{6.5cm}
2394 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2396 \begin{minipage}{6.5cm}
2397 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2400 \begin{minipage}{6.5cm}
2401 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2403 \begin{minipage}{6.5cm}
2405 \underline{Si-C bonds:}
2407 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2408 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2410 \underline{Si-Si bonds:}
2411 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2412 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2413 \underline{C-C bonds:}
2415 \item C-C next neighbour pairs reduced (mandatory)
2416 \item Peak at 0.3 nm slightly shifted
2418 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2419 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2421 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2423 \item Range [|-$\downarrow$]:
2424 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2425 with nearby Si$_{\text{I}}$}
2430 %\begin{picture}(0,0)(-330,-74)
2433 %\begin{minipage}{1.6cm}
2436 %stretched SiC\\[-0.1cm]
2443 \begin{pspicture}(0,0)(0,0)
2444 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2445 \begin{minipage}{10cm}
2447 {\color{blue}\bf Stretched SiC in c-Si}
2449 \item Consistent to precipitation model involving \cs{}
2450 \item Explains annealing behavior of high/low T C implants
2452 \item Low T: highly mobiel \ci{}
2453 \item High T: stable configurations of \cs{}
2456 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2457 $\Rightarrow$ Precipitation mechanism involving \cs{}
2467 Increased temperature simulations at high C concentration
2472 \begin{minipage}{6.5cm}
2473 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2475 \begin{minipage}{6.5cm}
2476 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2484 \begin{minipage}[t]{6.0cm}
2485 0.186 nm: Si-C pairs $\uparrow$\\
2486 (as expected in 3C-SiC)\\[0.2cm]
2487 0.282 nm: Si-C-C\\[0.2cm]
2488 $\approx$0.35 nm: C-Si-Si
2491 \begin{minipage}{0.2cm}
2495 \begin{minipage}[t]{6.0cm}
2496 0.15 nm: C-C pairs $\uparrow$\\
2497 (as expected in graphite/diamond)\\[0.2cm]
2498 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2499 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2504 \item Decreasing cut-off artifact
2505 \item {\color{red}Amorphous} SiC-like phase remains
2506 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2507 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2516 High C \& small $V$ \& short $t$
2519 Slow restructuring due to strong C-C bonds
2522 High C \& low T implants
2533 Summary and Conclusions
2541 \begin{minipage}[t]{12.9cm}
2542 \underline{Pecipitation simulations}
2544 \item High C concentration $\rightarrow$ amorphous SiC like phase
2545 \item Problem of potential enhanced slow phase space propagation
2546 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2547 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2548 \item High T necessary to simulate IBS conditions (far from equilibrium)
2549 \item Precipitation by successive agglomeration of \cs (epitaxy)
2550 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2551 (stretched SiC, interface)
2559 \begin{minipage}{12.9cm}
2564 \item Point defects excellently / fairly well described
2566 \item C$_{\text{sub}}$ drastically underestimated by EA
2567 \item EA predicts correct ground state:
2568 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2569 \item Identified migration path explaining
2570 diffusion and reorientation experiments by DFT
2571 \item EA fails to describe \ci{} migration:
2572 Wrong path \& overestimated barrier
2574 \item Combinations of defects
2576 \item Agglomeration of point defects energetically favorable
2577 by compensation of stress
2578 \item Formation of C-C unlikely
2579 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2580 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2581 Low barrier (\unit[0.77]{eV}) \& low capture radius
2589 \framebox{Precipitation by successive agglomeration of \cs{}}
2607 \underline{Augsburg}
2609 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2610 \item Ralf Utermann (EDV)
2613 \underline{Helsinki}
2615 \item Prof. K. Nordlund (MD)
2620 \item Bayerische Forschungsstiftung (financial support)
2623 \underline{Paderborn}
2625 \item Prof. J. Lindner (SiC)
2626 \item Prof. G. Schmidt (DFT + financial support)
2627 \item Dr. E. Rauls (DFT + SiC)
2628 \item Dr. S. Sanna (VASP)
2635 \bf Thank you for your attention!