1 \chapter{Molecular dynamics simulations}
3 The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures.
4 Both, quantum-mechanical and classical potential molecular dynamics simulations are performed.
5 Since quantum-mechanical calculations are restricted to a few hundreds of atoms only small volumes composed of three unit cells in each direction and small carbon concentrations are simulated using the VASP code.
6 Thus, investigations are restricted to the diffusion process of single carbon interstitials and the agglomeration of a few dumbbell interstitials in silicon.
7 Using classical potentials volume sizes up to 31 unit cells in each direction and high carbon concentrations are realizable.
8 Simulations targeting the formation of silicon carbide precipitates are, thus, attempted in classical potential calculations only.
10 \section{Ab initio MD simulations}
12 No pressure control, since VASP does not support this feature in MD mode.
13 The time step is set to one fs.
14 Explain some more parameters that differ from the latter calculations ...
16 Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.
18 \section{Classical potential MD simulations}
20 In contrast to the quantum-mechanical MD simulations the developed classical potential MD code is able to do constant pressure simulations using the Berendsen barostat.
21 The system pressure is set to zero pressure.
22 Due to promising advantages over the Tersoff potential the bond order potential of Erhard and Albe is used.
23 A time step of one fs is set.
25 \subsection{Simulations at temperatures used in ion beam synthesis}
26 \label{subsection:initial_sims}
28 In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized.
29 Periodic boundary conditions in each direction are applied.
30 The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved by ion beam synthesis (IBS).
31 After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted.
32 The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by
34 N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
35 =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
36 \label{eq:md:spheric_prec}
38 with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
39 In IBS experiments the smallest precipitates observed have radii starting from 2 nm up to 4 nm.
40 For the initial simulations a total amount of 6000 carbon atoms corresponding to a radius of approximately 3.1 nm is chosen.
41 In separated simulations these 6000 carbon atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell.
42 For reasons of simplification these regions are rectangularly shaped.
43 $V_1$ is chosen to be the total simulation volume.
44 $V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
45 $V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si.
46 The two latter insertion volumes are considered since no diffusion of carbon atoms is expected within the simulated period of time at prevalent temperatures.
47 {\color{red}Todo: Refere to diffusion simulations and Mattoni paper.}
48 For rectangularly shaped precipitates with side length $L$ the amount of carbon atoms in 3C-SiC and silicon atoms in c-Si is given by
50 N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
52 N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
55 Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting carbon concentrations inside these volumes.
56 Looking at the carbon concentrations simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of carbon.
59 \begin{tabular}{l c c c}
62 & $V_1$ & $V_2$ & $V_3$ \\
64 Side length [\AA] & 168.3 & 50.0 & 49.0 \\
65 Carbon concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
70 \caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 carbon atoms.}
71 \label{table:md:ins_vols}
74 The insertion is realized in a way to keep the system temperature constant.
75 In each of 600 insertion steps 10 carbon atoms are inserted at random positions within the respective region, which involves an increase in kinetic energy.
76 Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy.
77 Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures.
78 Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is $20\, ^{\circ}\mathrm{C}$ is reached.
79 Figure \ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
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153 \caption[Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process. $T_{\text{s}}$ and $p_{\text{s}}$ are the preset values for the system temperature and pressure. $T_{\text{avg}}$ is the averaged actual system temperature.}
154 \label{fig:md:prec_fc}
157 The radial distribution function $g(r)$ for C-C and Si-Si distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
160 \includegraphics[width=12cm]{sic_prec_450_si-si_c-c.ps}
162 \caption[Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature.]{Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature. The bright blue graph shows the Si-Si radial distribution for pure c-Si. The insets show magnified regions of the respective type of bond.}
163 \label{fig:md:pc_si-si_c-c}
167 \includegraphics[width=12cm]{sic_prec_450_energy.ps}
169 \caption[Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes.]{Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes. Arrows mark the end of carbon insertion and the start of the cooling process respectively.}
170 \label{fig:md:energy_450}
172 It is easily and instantly visible that there is no significant difference among the two simulations of high carbon concentration.
173 The first C-C peak appears at about 0.15 nm, which is compareable to the nearest neighbour distance of graphite or diamond.
174 The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume.
175 These carbon atoms are assumed to form strong bonds.
176 This is supported by figure \ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
177 A huge decrease of the total energy during carbon insertion is observed for the simulations with high carbon concentration in contrast to the $V_1$ simulation, which shows a slight increase.
178 The difference in energy $\Delta$ growing within the carbon insertion process up to a value of roughly 0.06 eV per atom persists unchanged until the end of the simulation.
180 The excess amount of next neighboured strongly bounded C-C bonds in the high concentration simulations make these configurations energetically more favorable compared to the low concentration configuration.
181 However, in the same way a lot of energy is needed to break these bonds to get out of the local energy minimum advancing towards the global minimum configuration.
182 Thus, such conformational chamges are very unlikely to happen.
183 This is in accordance with the constant total energy observed in the continuation step of 100 ps inbetween the end of carbon insertion and the cooling process.
184 Obviously no energetically favorable relaxation is taking place at a system temperature of $450\,^{\circ}\mathrm{C}$.
186 The C-C peak at about 0.31 nm perfectly matches the nearest neighbour distance of two carbon atoms in the 3C-SiC lattice.
187 As can be seen from the inset this peak is also observed for the $V_1$ simulation.
188 In 3C-SiC the same distance is also expected for nearest neighbour silicon atoms.
189 The bottom of figure \ref{fig:md:pc_si-si_c-c} shows the radial distribution of Si-Si bonds together with a reference graph for pure c-Si.
190 Indeed non-zero $g(r)$ values around 0.31 nm are observed while the amount of Si pairs at regular c-Si distances of 0.24 nm and 0.38 nm decreases.
191 However, no clear peak is observed but the interval of enhanced $g(r)$ values corresponds to the width of the C-C $g(r)$ peak.
192 In addition the abrupt increase of Si pairs at 0.29 nm can be attributed to the Si-Si cut-off radius of 0.296 nm as used in the present bond order potential.
193 The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
194 Without the abrubt increase a maximum around 0.31 nm gets even more conceivable.
195 For low concentrations of carbon, that is the $V_1$ simulation and early stages of the $V_2$ and $V_3$ simulations, analyses of configurations in which Si-Si distances around 0.3 nm appear and which are identifiable in regions of high disorder, which especially applies for the high concentration simulations, identify the \hkl<1 0 0> C-Si dumbbell to be responsible for stretching the Si-Si next neighbour distance.
196 This excellently agrees with the calculated value $r(13)$ in table \ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \hkl<1 0 0> C-Si dumbbell configuration.
200 \includegraphics[width=12cm]{sic_prec_450_si-c.ps}
202 \caption{Radial distribution function of the Si-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature together with Si-C bonds resulting in a C-Si \hkl<1 0 0> dumbbell configuration.}
203 \label{fig:md:pc_si-c}
205 Figure \ref{fig:md:pc_si-c} displays the Si-C radial distribution function for all three insertion volumes together with the Si-C bonds as observed in a C-Si \hkl<1 0 0> dumbbell configuration.
206 The first peak observed for all insertion volumes is at approximately 0.186 nm.
207 This corresponds quite well to the expected next neighbour distance of 0.189 nm for Si and C atoms in 3C-SiC.
208 By comparing the resulting Si-C bonds of a C-Si \hkl<1 0 0> dumbbell with the C-Si distances of the low concentration simulation it is evident that the resulting structure of the $V_1$ simulation is dominated by this type of defects.
209 This is not surpsising, since the \hkl<1 0 0> dumbbell is found to be the ground state defect of a C interstitial in c-Si and for the low concentration simulations a carbon interstitial is expected in every fifth silicon unit cell only, thus, excluding defect superposition phenomena.
210 The peak distance at 0.186 nm and the bump at 0.175 nm corresponds to the distance $r(3C)$ and $r(1C)$ as listed in table \ref{tab:defects:100db_cmp} and visualized in figure \ref{fig:defects:100db_cmp}.
211 In addition it can be easily identified that the \hkl<1 0 0> dumbbell configuration contributes to the peaks at about 0.335 nm, 0.386 nm, 0.434 nm, 0.469 nm and 0.546 nm observed in the $V_1$ simulation.
212 Not only the peak locations but also the peak widths and heights become comprehensible.
213 The distinct peak at 0.26 nm, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
215 For high carbon concentrations, that is the $V_2$ and $V_3$ simulation, the defect concentration is likewiese increased and a considerable amount of damage is introduced in the insertion volume.
216 The consequential superposition of these defects and the high amounts of damage generate new displacement arrangements for the C-C as well as for the Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
217 Short range order indeed is observed but only hardly visible is the long range order.
218 This indicates the formation of an amorphous SiC-like phase.
219 In fact the resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential \cite{gao02}.
221 \subsection{Limitations of conventional MD and short range potentials}
223 At first the formation of an amorphous SiC-like phase is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
224 On closer inspection, however, reasons become clear, which are discussed in the following.
226 The first reason is a general problem of MD simulations in conjunction with limitations in computer power, which results in a slow and restricted propagation in phase space.
227 In molecular systems, characteristic motions take place over a wide range of time scales.
228 Vibrations of the covalent bond take place on the order of $10^{-14}\,\text{s}$ of which the thermodynamic and kinetic properties are well described by MD simulations.
229 To avoid dicretization errors the integration timestep needs to be chosen smaller than the fastest vibrational frequency in the system.
230 On the other hand, infrequent processes, such as conformational changes, reorganization processes during film growth, defect diffusion and phase transitions are processes undergoing long-term evolution in the range of microseconds.
231 This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, that is the system temperature.
232 Thus, the average time of a transition from one potential basin to another corresponds to a great deal of vibrational periods, which in turn determine the integration timestep.
233 Hence, time scales covering the neccessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
234 New methods have been developed to bypass the time scale problem like hyperdnyamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature acclerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99} retaining proper thermodynmic sampling.
236 In addition to the time scale limitation, problems attributed to the short range potential exist.
237 The sharp cut-off funtion, which limits the interacting ions to the next neighboured atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbour distance, is responsible for overestimated and unphysical high forces of next neighboured atoms \cite{tang95,mattoni2007}.
238 Indeed it is not only the strong C-C bond which is hard to break inhibiting carbon diffusion and further rearrengements.
239 This is also true for the low concentration simulations dominated by the occurrence of C-Si dumbbells spread over the whole simulation volume.
240 The bonds of these C-Si pairs are also affected by the cut-off artifact preventing carbon diffusion and agglomeration of the dumbbells.
241 This can be seen from the almost horizontal progress of the total energy graph in the continuation step, even for the low concentration simulation.
242 The unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
243 However, since valueable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
244 One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature \cite{mattoni2007}.
245 Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC \cite{tang95} but might be rather ineffective for the challange inherent to this study.
247 To conclude the obstacle needed to get passed is twofold.
248 The sharp cut-off of the used bond order model potential introduces overestimated high forces between next neighboured atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
249 Thus, pushing the time scale to the limits of computational ressources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
251 Instead the first approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
252 Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock \cite{sorensen2000}.
253 The TAD corrections are not applied in coming up simulations.
254 This is justified by two reasons.
255 First of all a compensation of the overestimated bond strengthes due to the short range potential is expected.
256 Secondly there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than $450\,^{\circ}\mathrm{C}$ in IBS \cite{lindner01}.
257 It is therefore expected that the kinetics affecting the 3C-SiC precipitation are not much different at higher temperatures aside from the fact that it is occuring much more faster.
258 Moreover, the interest of this study is focused on structural evolution of a system far from equilibrium instead of equilibrium properties which rely upon proper phase space sampling.
259 On the other hand, during implantation, the actual temperature inside the implantation volume is definetly higher than the experimentally determined temperature tapped from the surface of the sample.
261 \subsection{Increased temperature simulations}
263 Due to the limitations of short range potentials and conventional MD as discussed above elevated temperatures are used in the following.
264 The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection:initial_sims}.
265 Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and refered to as low carbon and high carbon concentration simulations.
266 Temperatures ranging from $450\,^{\circ}\mathrm{C}$ up to $2050\,^{\circ}\mathrm{C}$ are used.
267 A quality value $Q$ is introduced, which is defined as
269 Q = \frac{\text{Amount of C atoms with 4 next neighboured Si atoms}}
270 {\text{Total amount of C atoms}} \text{ .}
273 In 3C-SiC every C atom has four next neighboured Si atoms resulting in $Q=1$.
275 Figure ... shows the radial distribution of Si-C bonds and the corresponding quality paragraphs.
277 \subsection{Constructed 3C-SiC precipitate in crystalline silicon}
279 {\color{red}Todo: We want to know where we want to go ...}
281 In the following a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed as it is expected from IBS experiments and from simulations that finally succeed simulating the precipitation event.
282 On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of energy for configurations that are aimed to be reproduced by simulation possibly enabling the prediction of conditions necessary for the simulation of the precipitation process.
283 On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, investigations of the behaviour of such precipitates and the surrounding can be made.
285 To construct a spherical 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
286 A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is used.
287 To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
288 The initial precipitate configuration is constructed in two steps.
289 In the first step the surrounding silicon matrix is created.
290 This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
291 The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
292 In a second step 3C-SiC is created inside the empty sphere of radius $x$.
293 The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated.
294 This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
296 \frac{8}{a_{\text{Si}}^3}(
297 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
298 -\frac{4}{3}\pi x^3)+
299 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
301 \label{eq:md:constr_sic_01}
304 which can be simplified to read
306 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
307 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
308 \label{eq:md:constr_sic_02}
312 %x^3=\frac{16\pi}{5500 \cdot 3}y^3=
313 %\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
315 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
317 \label{eq:md:constr_sic_03}
319 By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
320 Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
321 However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
324 \begin{tabular}{l c c c c}
327 & C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
329 Expected & 5500 & 5500 & 68588 & 74088\\
330 Obtained & 5495 & 5486 & 68591 & 74077\\
331 Difference & 5 & 14 & -3 & 11\\
335 \caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.}
336 \label{table:md:sic_prec}
340 After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
341 Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
342 Once the main part of th excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
344 PC and energy of that one.
346 Now let's see, whether annealing will lead to some energetically more favorable configurations.
348 Estimate surface energy ...
350 \subsection{Simulations at temperatures exceeding the silicon melting point}
352 LL Cool J is hot as hell!
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