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5 \usepackage[english,german]{babel}
9 \selectlanguage{english}
11 \renewcommand\labelitemii{{\color{black}$\bullet$}}
15 % Fliessenden Hintergrund von RGB-Farbe 1. .98 .98 nach 1. .85 .85
16 % und wieder nach 1. .98 .98 (1. .85 .85 wird nach 0.1=10% des Hinter-
18 % Achtung Werte unter .8 verbrauchen zu viel Tinte!!!
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24 \newrgbcolor{si-yellow}{.6 .6 0}
26 % Groesse der einzelnen Spalten als Anteil der Gesamt-Textbreite
27 \renewcommand{\columnfrac}{.31}
30 \newcommand{\pot}{\mathcal{V}}
35 \centerline{{\Huge \bfseries Molecular dynamics simulation
36 of defect formation and precipitation}}
38 \centerline{{\Huge \bfseries in heavily carbon doped silicon}}
40 \centerline{\huge\textsc {\underline{F.~Zirkelbach}$^1$,
42 K.~Nordlund$^2$, B.~Stritzker$^1$}}
45 \begin{minipage}{.065\textwidth}
46 \includegraphics[height=5.5cm]{uni-logo.eps}
48 \begin{minipage}{.57\textwidth}
49 \centerline{\Large $^1$ Experimentalphysik IV, Institut f\"ur Physik,
50 Universit\"at Augsburg,}
51 \centerline{\Large Universit\"atsstr. 1, D-86135 Augsburg, Germany}
53 \begin{minipage} {.065\textwidth}
54 \includegraphics[height=5cm]{Lehrstuhl-Logo.eps}
58 \begin{minipage}{.20\textwidth}
59 \includegraphics[height=5.5cm]{logo_eng.eps}
61 \begin{minipage}{.50\textwidth}
62 \centerline{\Large $^2$ Accelerator Laboratory,
63 Department of Physical Sciences,
64 University of Helsinki,}
65 \centerline{\Large Pietari Kalmink. 2, 00014 Helsinki, Finland}
76 {\bf Reasons for understanding the 3C-SiC precipitation process}
78 \item Significant technological progress
79 in 3C-SiC wide band gap semiconductor thin film formation [1].
80 \item New perspectives for processes relying upon prevention of
81 precipitation, e.g. fabrication of strained pseudomorphic
82 $\text{Si}_{1-y}\text{C}_y$ heterostructures [2].
85 [1] J. H. Edgar, J. Mater. Res. 7 (1992) 235.}\\
87 [2] J. W. Strane, S. R. Lee, H. J. Stein, S. T. Picraux,
88 J. K. Watanabe, J. W. Mayer, J. Appl. Phys. 79 (1996) 637.}
91 \section*{Crystalline silicon and cubic silicon carbide}
92 {\bf Lattice types and unit cells:}
94 \item Crystalline silicon (c-Si) has diamond structure\\
95 $\Rightarrow {\color{si-yellow}\bullet}$ and
96 ${\color{gray}\bullet}$ are Si atoms
97 \item Cubic silicon carbide (3C-SiC) has zincblende structure\\
98 $\Rightarrow {\color{si-yellow}\bullet}$ are Si atoms,
99 ${\color{gray}\bullet}$ are C atoms
101 \begin{minipage}{15cm}
102 {\bf Lattice constants:}
104 4a_{\text{c-Si}}\approx5a_{\text{3C-SiC}}
106 {\bf Silicon density:}
108 \frac{n_{\text{3C-SiC}}}{n_{\text{c-Si}}}=97,66\,\%
111 \begin{minipage}{10cm}
112 \includegraphics[width=10cm]{sic_unit_cell.eps}
116 \section*{Supposed Si to 3C-SiC conversion}
117 {\bf Schematic of the conversion mechanism}\\\\
118 \begin{minipage}{7.8cm}
119 \includegraphics[width=7.7cm]{sic_prec_seq_01.eps}
122 \begin{minipage}{7.8cm}
123 \includegraphics[width=7.7cm]{sic_prec_seq_02.eps}
126 \begin{minipage}{7.8cm}
127 \includegraphics[width=7.7cm]{sic_prec_seq_03.eps}
131 \item Formation of C-Si dumbbells on regular c-Si lattice sites
132 \item Agglomeration into large clusters (embryos)
133 \item Precipitation of 3C-SiC + Creation of interstitials
136 {\bf Experimental observations} [3]
138 \item Minimal diameter of precipitation: 2 - 4 nm
139 \item Equal orientation of c-Si and 3C-SiC (hkl)-planes
142 [3] J. K. N. Lindner, Appl. Phys. A 77 (2003) 27.
146 \section*{Simulation details}
149 \item Microscopic description of N particles
150 \item Analytical interaction potential
151 \item Propagation rule in 6N-dim. phase space:
152 Hamilton's equations of motion
153 \item Observables obtained by time or ensemble averages
155 {\bf Application details:}\\[0.5cm]
156 \begin{minipage}{17cm}
158 \item Integrator: Velocity Verlet, timestep: 1 fs
159 \item Ensemble: isothermal-isobaric NPT [4]
161 \item Berendsen thermostat:
162 $\tau_{\text{T}}=100\text{ fs}$
163 \item Brendsen barostat:\\
164 $\tau_{\text{P}}=100\text{ fs}$,
165 $\beta^{-1}=100\text{ GPa}$
167 \item Potential: Tersoff-like bond order potential [5]
169 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
170 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
174 \begin{minipage}{9cm}
175 \includegraphics[width=9cm]{tersoff_angle.eps}
176 \end{minipage}\\[1cm]
178 [4] L. Verlet, Phys. Rev. 159 (1967) 98.}\\
180 [5] P. Erhart and K. Albe, Phys. Rev. B 71 (2005) 35211.}
187 \section*{Interstitial configurations}
188 {\bf Simulation sequence:}\\
190 \begin{minipage}{15cm}
192 \begin{pspicture}(0,0)(14,14)
193 \rput(7,12.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
196 \item Initial configuration: $9\times9\times9$ unit cells Si
197 \item Periodic boundary conditions
198 \item $T=0\text{ K}$, $p=0\text{ bar}$
201 \rput(7,6){\rnode{insert}{\psframebox{
203 Insertion of C / Si atom:
205 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
206 (${\color{red}\triangleleft}$)
207 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
208 (${\color{green}\triangleright}$)
209 \item $(-1/8,-1/8,-1/4)$, $(-3/8,-3/8,-1/4)$\\
210 $\rightarrow$ {\color{magenta}110 dumbbell}
211 (${\color{magenta}\Box}$,$\circ$)
212 \item random positions (critical distance check)
215 \rput(7,1.5){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
217 Relaxation time: 2 ps
219 \ncline[]{->}{init}{insert}
220 \ncline[]{->}{insert}{cool}
224 \begin{minipage}{10cm}
225 \includegraphics[width=11cm]{unit_cell_s.eps}
228 {\bf Si self-interstitial results:}\\
231 \begin{minipage}[t]{8.5cm}
232 \underline{Tetrahedral}\\
234 \includegraphics[width=8cm]{si_self_int_tetra_0.eps}
236 \begin{minipage}[t]{8.5cm}
237 \underline{110 dumbbell}\\
239 \includegraphics[width=8cm]{si_self_int_dumbbell_0.eps}
241 \begin{minipage}[t]{8.5cm}
242 \underline{Hexagonal}\\
243 $E_f^{\star}\approx4.48$ eV (unstable!)\\
244 \includegraphics[width=8cm]{si_self_int_hexa_0.eps}
245 \end{minipage}\\[1cm]
247 \underline{Random insertion}\\
249 \begin{minipage}{8.5cm}
251 \includegraphics[width=8cm]{si_self_int_rand_397_0.eps}
253 \begin{minipage}{8.5cm}
255 \includegraphics[width=8cm]{si_self_int_rand_375_0.eps}
257 \begin{minipage}{8.5cm}
259 \includegraphics[width=8cm]{si_self_int_rand_356_0.eps}
260 \end{minipage}\\[1cm]
263 {\bf C in Si interstitial results:}\\
266 \begin{minipage}[t]{8.5cm}
267 \underline{Tetrahedral}\\
269 \includegraphics[width=8cm]{c_in_si_int_tetra_0.eps}
271 \begin{minipage}[t]{8.5cm}
272 \underline{110 dumbbell}\\
274 \includegraphics[width=8cm]{c_in_si_int_dumbbell_0.eps}
276 \begin{minipage}[t]{8.5cm}
277 \underline{Hexagonal}\\
278 $E_f^{\star}\approx5.6$ eV (unstable!)\\
279 \includegraphics[width=8cm]{c_in_si_int_hexa_0.eps}
280 \end{minipage}\\[1cm]
282 \begin{minipage}{17cm}
283 \underline{\flq100\frq{} dumbbell configuration}
286 \item Very often observed
287 \item Most energetically favorable configuration
288 \item Experimental evidence [6]
291 \begin{minipage}{8cm}
292 \includegraphics[width=8cm]{c_in_si_int_001db_0.eps}
293 \end{minipage}\\[1cm]
295 \includegraphics[width=24cm]{100-c-si-db_s.eps}
298 [6] G. D. Watkins and K. L. Brower, Phys. Rev. Lett. 36 (1976) 1329.}
306 \section*{High C concentration simulations}
308 {\bf Simulation sequence:}\\
311 \begin{pspicture}(0,0)(30,13)
313 \rput(7.5,11){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
316 \item Initial configuration: $31\times31\times31$ unit cells Si
317 \item Periodic boundary conditions
318 \item $T=450\, ^{\circ}\textrm{C}$, $p=0\text{ bar}$
319 \item Equilibration of $E_{kin}$ and $E_{pot}$
322 \rput(7.5,5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
324 Insertion of 6000 carbon atoms at constant\\
327 \item Total simulation volume $V_1$
328 \item Volume of minimal 3C-SiC precipitation $V_2$
329 \item Volume of necessary amount of Si $V_3$
332 \rput(7.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
334 Cooling down to $20\, ^{\circ}\textrm{C}$
336 \ncline[]{->}{init}{insert}
337 \ncline[]{->}{insert}{cool}
338 \psframe[fillstyle=solid,fillcolor=white](16,2.6)(26,12.6)
339 \psframe[fillstyle=solid,fillcolor=lightgray](18,4.6)(24,10.6)
340 \psframe[fillstyle=solid,fillcolor=gray](18.5,5.1)(23.5,10.1)
341 \rput(9,5.4){\pnode{in1}}
342 \rput(15,5.4){\pnode{in-1}}
343 \rput(17,7.2){\pnode{ins1}}
344 \rput(14,4.2){\pnode{in2}}
345 \rput(15,4.2){\pnode{in-2}}
346 \rput(18.25,6.88){\pnode{ins2}}
347 \rput(12,3.0){\pnode{in3}}
348 \rput(15,3.0){\pnode{in-3}}
349 \rput(21,7.6){\pnode{ins3}}
350 \ncline[linewidth=0.05]{->}{in-1}{ins1}
351 \ncline[linewidth=0.05]{->}{in-2}{ins2}
352 \ncline[linewidth=0.05]{->}{in-3}{ins3}
353 \ncline[linewidth=0.05]{-}{in1}{in-1}
354 \ncline[linewidth=0.05]{-}{in2}{in-2}
355 \ncline[linewidth=0.05]{-}{in3}{in-3}
359 Si-C and C-C pair correlation function:\\
360 \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-c_c-c.eps}
363 {\bf Dashed vertical lines:} Further calculated C-Si distances
364 in the \flq100\frq{} C-Si dumbbell interstitial configuration}\\[0.5cm]
366 Si-Si pair correlation function:\\
367 \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-si.eps}\\
368 {\bf Interpretation:}
371 \item C-C peak at 0.15 nm similar to next neighbour distance of graphite
373 $\Rightarrow$ Formation of strong C-C bonds
374 (almost only for high C concentrations)
375 \item C-C peak at 0.31 nm equals C-C distance in 3C-SiC\\
376 (due to concatenated, differently oriented
377 \flq100\frq{} dumbbell interstitials)
378 \item Si-Si shows non-zero g(r) values around 0.31 nm
379 and a decrease at regular distances\\
381 interval of enhanced g(r) corresponds to C-C peak width)
382 \item Si-C peak at 0.19 nm similar to next neighbour distance in 3C-SiC
383 \item Low C concentration (i.e. $V_1$):
384 The \flq100\frq{} dumbbell configuration
386 \item is identified to stretch the Si-Si next neighbour distance
388 \item is identified to contribute to the Si-C peak at 0.19 nm
389 \item explains further C-Si peaks (dashed vertical lines)
391 $\Rightarrow$ C atoms are first elements arranged at distances
392 expected for 3C-SiC\\
393 $\Rightarrow$ C atoms pull the Si atoms into the right
394 configuration at a later stage
395 \item High C concentration (i.e. $V_2$ and $V_3$):
397 \item High amount of damage introduced into the system
398 \item Short range order observed but almost no long range order
400 $\Rightarrow$ Start of amorphous SiC-like phase formation\\
401 $\Rightarrow$ Higher temperatures required for proper SiC formation
408 \section*{Conclusion}
410 \item \flq100\frq{} C-Si dumbbell interstitial configuration is observed
411 to be the energetically most favorable configuration
412 \item For low C concentrations C atoms introduced as differently
413 oriented C-Si dumbbells in c-Si are properly arranged
415 \item For high C concentrations an amorphous SiC-like phase is observed
416 which suggests higher temperature simulation runs for proper