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18 % Achtung Werte unter .8 verbrauchen zu viel Tinte!!!
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29 % Groesse der einzelnen Spalten als Anteil der Gesamt-Textbreite
30 \renewcommand{\columnfrac}{.31}
33 \newcommand{\pot}{\mathcal{V}}
38 \centerline{{\Huge \bfseries Molecular dynamics simulation
39 of defect formation and precipitation}}
41 \centerline{{\Huge \bfseries in heavily carbon doped silicon}}
43 \centerline{\huge\textsc {\underline{F.~Zirkelbach}$^1$,
45 K.~Nordlund$^2$, B.~Stritzker$^1$}}
48 \begin{minipage}{.065\textwidth}
49 \includegraphics[height=5.5cm]{uni-logo.eps}
51 \begin{minipage}{.57\textwidth}
52 \centerline{\Large $^1$ Experimentalphysik IV, Institut f\"ur Physik,
53 Universit\"at Augsburg,}
54 \centerline{\Large Universit\"atsstr. 1, D-86135 Augsburg, Germany}
56 \begin{minipage} {.065\textwidth}
57 \includegraphics[height=5cm]{Lehrstuhl-Logo.eps}
61 \begin{minipage}{.20\textwidth}
62 \includegraphics[height=5.5cm]{logo_eng.eps}
64 \begin{minipage}{.50\textwidth}
65 \centerline{\Large $^2$ Accelerator Laboratory,
66 Department of Physical Sciences,
67 University of Helsinki,}
68 \centerline{\Large Pietari Kalmink. 2, 00014 Helsinki, Finland}
79 {\bf Importance of the 3C-SiC precipitation process in silicon}
81 \item SiC is a promising wide band gap material for high-temperature,
82 high-power, high-frequency semiconductor devices [1].
83 \item 3C-SiC epitaxial thin film formation on Si requires detailed
84 knowledge of SiC nucleation.
85 \item Fabrication of high carbon doped, strained pseudomorphic
86 $\text{Si}_{1-y}\text{C}_y$ layers requires suppression of
87 3C-SiC nucleation [2].
90 [1] J. H. Edgar, J. Mater. Res. 7 (1992) 235.}\\
92 [2] J. W. Strane, S. R. Lee, H. J. Stein, S. T. Picraux,
93 J. K. Watanabe, J. W. Mayer, J. Appl. Phys. 79 (1996) 637.}
97 \section*{Crystalline silicon and cubic silicon carbide}
98 {\bf Lattice types and unit cells:}
100 \item Crystalline silicon (c-Si) has diamond structure\\
101 $\Rightarrow {\color{si-yellow}\bullet}$ and
102 ${\color{gray}\bullet}$ are Si atoms
103 \item Cubic silicon carbide (3C-SiC) has zincblende structure\\
104 $\Rightarrow {\color{si-yellow}\bullet}$ are Si atoms,
105 ${\color{gray}\bullet}$ are C atoms
107 \begin{minipage}{15cm}
108 {\bf Lattice constants:}
110 4a_{\text{c-Si}}\approx5a_{\text{3C-SiC}}
112 {\bf Silicon density:}
114 \frac{n_{\text{3C-SiC}}}{n_{\text{c-Si}}}=97,66\,\%
117 \begin{minipage}{10cm}
118 \includegraphics[width=10cm]{sic_unit_cell.eps}
123 \section*{Supposed Si to 3C-SiC conversion}
124 {\bf Schematic of the conversion mechanism}\\\\
125 \begin{minipage}[c]{8.8cm}
126 \includegraphics[width=8.0cm]{sic_prec_seq_01.eps}
128 \begin{minipage}[c]{8.8cm}
129 \includegraphics[width=8.0cm]{sic_prec_seq_02.eps}
131 \begin{minipage}[c]{8.1cm}
132 \includegraphics[width=8.0cm]{sic_prec_seq_03.eps}
136 \item Formation of C-Si dumbbells on regular c-Si lattice sites
137 \item Agglomeration into large clusters (embryos)
138 \item Precipitation of 3C-SiC + Creation of interstitials
141 {\bf Experimental observations} [3]
143 \item Minimal radius of precipitates: 2 - 4 nm
144 \item Equal orientation of c-Si and 3C-SiC (hkl)-planes
147 [3] J. K. N. Lindner, Appl. Phys. A 77 (2003) 27.
152 \section*{Simulation details}
155 \item Microscopic description of N particles
156 \item Analytical interaction potential
157 \item Propagation rule in 6N-dim. phase space:
158 Hamilton's equations of motion
159 \item Observables obtained by time or ensemble averages
161 {\bf Application details:}\\[0.5cm]
162 \begin{minipage}{17cm}
164 \item Integrator: Velocity Verlet, timestep: 1 fs
165 \item Ensemble: isothermal-isobaric NPT [4]
167 \item Berendsen thermostat:
168 $\tau_{\text{T}}=100\text{ fs}$
169 \item Brendsen barostat:\\
170 $\tau_{\text{P}}=100\text{ fs}$,
171 $\beta^{-1}=100\text{ GPa}$
173 \item Potential: Tersoff-like bond order potential [5]
175 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
176 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
180 \begin{minipage}{9cm}
181 \includegraphics[width=9cm]{tersoff_angle.eps}
182 \end{minipage}\\[1cm]
184 [4] L. Verlet, Phys. Rev. 159 (1967) 98.}\\
186 [5] P. Erhart and K. Albe, Phys. Rev. B 71 (2005) 35211.}
193 \section*{Interstitial configurations}
194 {\bf Simulation sequence:}\\
196 \begin{minipage}{15cm}
198 \begin{pspicture}(0,0)(14,14)
199 \rput(7,12.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
202 \item Initial configuration: $9\times9\times9$ unit cells Si
203 \item Periodic boundary conditions
204 \item $T=0\text{ K}$, $p=0\text{ bar}$
207 \rput(7,6){\rnode{insert}{\psframebox{
209 Insertion of C / Si atom:
211 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
212 (${\color{red}\triangleleft}$)
213 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
214 (${\color{green}\triangleright}$)
215 \item $(-1/8,-1/8,-1/4)$, $(-3/8,-3/8,-1/4)$\\
216 $\rightarrow$ {\color{magenta}110 dumbbell}
217 (${\color{magenta}\Box}$,$\circ$)
218 \item random positions (critical distance check)
221 \rput(7,1.5){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
223 Relaxation time: 2 ps
225 \ncline[]{->}{init}{insert}
226 \ncline[]{->}{insert}{cool}
230 \begin{minipage}{10cm}
231 \includegraphics[width=11cm]{unit_cell_s.eps}
234 {\bf Si self-interstitial results:}\\
237 \begin{minipage}[t]{8.5cm}
238 \underline{Tetrahedral}\\
240 \includegraphics[width=8cm]{si_self_int_tetra_0.eps}
242 \begin{minipage}[t]{8.5cm}
243 \underline{110 dumbbell}\\
245 \includegraphics[width=8cm]{si_self_int_dumbbell_0.eps}
247 \begin{minipage}[t]{8.5cm}
248 \underline{Hexagonal}\\
249 $E_f^{\star}\approx4.48$ eV (unstable!)\\
250 \includegraphics[width=8cm]{si_self_int_hexa_0.eps}
251 \end{minipage}\\[1cm]
253 \underline{Random insertion}\\
255 \begin{minipage}{8.5cm}
257 \includegraphics[width=8cm]{si_self_int_rand_397_0.eps}
259 \begin{minipage}{8.5cm}
261 \includegraphics[width=8cm]{si_self_int_rand_375_0.eps}
263 \begin{minipage}{8.5cm}
265 \includegraphics[width=8cm]{si_self_int_rand_356_0.eps}
266 \end{minipage}\\[1cm]
269 {\bf C in Si interstitial results:}\\
272 \begin{minipage}[t]{8.5cm}
273 \underline{Tetrahedral}\\
275 \includegraphics[width=8cm]{c_in_si_int_tetra_0.eps}
277 \begin{minipage}[t]{8.5cm}
278 \underline{110 dumbbell}\\
280 \includegraphics[width=8cm]{c_in_si_int_dumbbell_0.eps}
282 \begin{minipage}[t]{8.5cm}
283 \underline{Hexagonal}\\
284 $E_f^{\star}\approx5.6$ eV (unstable!)\\
285 \includegraphics[width=8cm]{c_in_si_int_hexa_0.eps}
286 \end{minipage}\\[1cm]
288 \begin{minipage}{17cm}
289 \underline{\flq100\frq{} dumbbell configuration}
292 \item Very often observed
293 \item Most energetically favorable configuration
294 \item Experimental evidence [6]
297 \begin{minipage}{8cm}
298 \includegraphics[width=8cm]{c_in_si_int_001db_0.eps}
299 \end{minipage}\\[1cm]
301 \includegraphics[width=26cm]{100-c-si-db_s.eps}\\[0.35cm]
304 [6] G. D. Watkins and K. L. Brower, Phys. Rev. Lett. 36 (1976) 1329.}
312 \section*{High C concentration simulations}
314 {\bf Simulation sequence:}\\
317 \begin{pspicture}(0,0)(30,13)
319 \rput(7.5,11){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
322 \item Initial configuration: $31\times31\times31$ unit cells Si
323 \item Periodic boundary conditions
324 \item $T=450\, ^{\circ}\textrm{C}$, $p=0\text{ bar}$
325 \item Equilibration of $E_{kin}$ and $E_{pot}$
328 \rput(7.5,5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
330 Insertion of 6000 carbon atoms at constant\\
331 temperature into $V_1$ or $V_2$ or $V_3$:
333 \item Total simulation volume $V_1$
334 \item Volume of minimal 3C-SiC precipitation $V_2$
335 \item Volume of necessary amount of Si $V_3$
338 \rput(7.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
340 Cooling down to $20\, ^{\circ}\textrm{C}$
342 \ncline[]{->}{init}{insert}
343 \ncline[]{->}{insert}{cool}
344 \psframe[fillstyle=solid,fillcolor=white](16,2.6)(26,12.6)
345 \psframe[fillstyle=solid,fillcolor=lightgray](18,4.6)(24,10.6)
346 \psframe[fillstyle=solid,fillcolor=gray](18.5,5.1)(23.5,10.1)
347 \rput(9,5.4){\pnode{in1}}
348 \rput(15,5.4){\pnode{in-1}}
349 \rput(17,7.2){\pnode{ins1}}
350 \rput(14,4.2){\pnode{in2}}
351 \rput(15,4.2){\pnode{in-2}}
352 \rput(18.25,6.88){\pnode{ins2}}
353 \rput(12,3.0){\pnode{in3}}
354 \rput(15,3.0){\pnode{in-3}}
355 \rput(21,7.6){\pnode{ins3}}
356 \ncline[linewidth=0.05]{->}{in-1}{ins1}
357 \ncline[linewidth=0.05]{->}{in-2}{ins2}
358 \ncline[linewidth=0.05]{->}{in-3}{ins3}
359 \ncline[linewidth=0.05]{-}{in1}{in-1}
360 \ncline[linewidth=0.05]{-}{in2}{in-2}
361 \ncline[linewidth=0.05]{-}{in3}{in-3}
365 Si-C and C-C pair correlation function:\\
366 \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-c_c-c.eps}
369 {\bf Dashed vertical lines:} Further calculated C-Si distances
370 in the \flq100\frq{} C-Si dumbbell interstitial configuration}\\[0.5cm]
372 Si-Si pair correlation function:\\
373 \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-si.eps}\\
374 {\bf Interpretation:}
377 \item C-C peak at 0.15 nm similar to next neighbour distance of graphite
379 $\Rightarrow$ Formation of strong C-C bonds
380 (almost only for high C concentrations)
381 \item Si-C peak at 0.19 nm similar to next neighbour distance in 3C-SiC
382 \item C-C peak at 0.31 nm equals C-C distance in 3C-SiC\\
383 (due to concatenated, differently oriented
384 \flq100\frq{} dumbbell interstitials)
385 \item Si-Si shows non-zero g(r) values around 0.31 nm like in 3C-SiC\\
386 and a decrease at regular distances\\
388 interval of enhanced g(r) corresponds to C-C peak width)
389 \item Low C concentration (i.e. $V_1$):
390 The \flq100\frq{} dumbbell configuration
392 \item is identified to stretch the Si-Si next neighbour distance
394 \item is identified to contribute to the Si-C peak at 0.19 nm
395 \item explains further C-Si peaks (dashed vertical lines)
397 $\Rightarrow$ C atoms are first elements arranged at distances
398 expected for 3C-SiC\\
399 $\Rightarrow$ C atoms pull the Si atoms into the right
400 configuration at a later stage
401 \item High C concentration (i.e. $V_2$ and $V_3$):
403 \item High amount of damage introduced into the system
404 \item Short range order observed but almost no long range order
406 $\Rightarrow$ Start of amorphous SiC-like phase formation\\
407 $\Rightarrow$ Higher temperatures required for proper SiC formation
414 \section*{Conclusion}
416 \item \flq100\frq{} C-Si dumbbell interstitial configuration is observed
417 to be the energetically most favorable configuration
418 \item For low C concentrations C atoms introduced as differently
419 oriented C-Si dumbbells in c-Si are properly arranged
421 \item For high C concentrations an amorphous SiC-like phase is observed
422 which suggests higher temperature simulation runs for proper
428 One of us (F. Z.) wants to acknowledge financial support by the\\
429 {\bf Bayerische Forschungsstiftung} (DPA-61/05).