1 \documentclass[prb,twocolumn,superscriptaddress,a4paper,showkeys,showpacs]{revtex4}
16 % ref mod: no capital letters (by editors)
17 \title{Combined ab initio and classical potential simulation study on the silicon carbide precipitation in silicon}
18 \author{F. Zirkelbach}
20 \affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}
22 \affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}
23 \author{J. K. N. Lindner}
24 \author{W. G. Schmidt}
26 \affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
29 Atomistic simulations on the silicon carbide precipitation in bulk silicon employing both, classical potential and first-principles methods are presented.
30 For the quantum-mechanical treatment basic processes assumed in the precipitation process are calculated in feasible systems of small size.
31 Results of the accurate first-principles calculations on defects and carbon diffusion in silicon are compared to results of classical potential simulations revealing significant limitations of the latter method.
32 An approach to work around this problem is proposed.
33 Finally results of the classical potential molecular dynamics simulations of large systems are discussed.
36 \keywords{point defects, migration, interstitials, first-principles calculations, classical potentials, molecular dynamics, silicon carbide, ion implantation}
37 \pacs{61.72.J-,61.72.-y,66.30.Lw,66.30.-h,31.15.A-,31.15.xv,34.20.Cf,61.72.uf}
40 % --------------------------------------------------------------------------------
41 \section{Introduction}
43 % TOOD: redo complete intro!
45 The wide band gap semiconductor silicon carbide (SiC) is well known for its outstanding physical and chemical properties.
46 The high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with the unique thermal and mechanical stability as well as radiation hardness makes SiC a suitable material for high-temperature, high-frequency and high-power devices operational in harsh and radiation-hard environments\cite{edgar92,morkoc94,wesch96,capano97,park98}.
47 Different modifications of SiC exist, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
48 Different polytypes exhibit different properties, in which the cubic phase of SiC (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is, thus, most effective for high-performance electronic devices.
49 Much progress has been made in 3C-SiC thin film growth by chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) on hexagonal SiC\cite{powell90,fissel95,fissel95_apl} and Si\cite{nishino83,nishino87,kitabatake93,fissel95_apl} substrates.
50 However, the frequent occurrence of defects such as twins, dislocations and double position boundaries remains a challenging problem.
51 Apart from these methods, high-dose carbon implantation into crystalline silicon (c-Si) with subsequent or in situ annealing was found to result in SiC microcrystallites in Si\cite{borders71}.
52 Utilized and enhanced, ion beam synthesis (IBS) has become a promising method to form thin SiC layers of high quality and exclusively of the 3C polytype embedded in and epitactically aligned to the Si host featuring a sharp interface\cite{lindner99,lindner01,lindner02}.
54 However, only little is known about the SiC conversion in C implanted Si.
57 \subfigure[]{\label{fig:hrem:c-si}\includegraphics[width=0.48\columnwidth]{../img/tem_c-si-db.eps}}
58 \subfigure[]{\label{fig:hrem:sic}\includegraphics[width=0.48\columnwidth]{../img/tem_3c-sic.eps}}
60 \caption{High resolution transmission electron microscopy (HREM) micrographs\cite{lindner99_2} of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes (a) and equally sized Moir\'e patterns indicating 3C-SiC precipitates (b).}
63 High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:hrem:c-si}.
64 A topotactic transformation into a 3C-SiC precipitate occurs once a critical radius of 2 nm to 4 nm is reached, which is manifested by the disappearance of the dark contrasts in favor of Moir\'e patterns (Fig.~\ref{fig:hrem:sic}) due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
65 The insignificantly lower Si density of SiC ($\approx \unit[4]{\%}$) compared to c-Si results in the emission of only a few excess Si atoms.
66 In contrast, investigations of strained Si$_{1-y}$C$_y$/Si heterostructures formed by MBE\cite{strane94,guedj98}, which incidentally involve the formation of SiC nanocrystallites, suggest an initial coherent precipitation by agglomeration of substitutional instead of interstitial C.
67 Coherency is lost once the increasing strain energy of the stretched SiC structure surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the Si substrate.
68 These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
69 While in CVD and MBE surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal.
70 However, in another IBS study Nejim et~al.\cite{nejim95} propose a topotactic transformation that is likewise based on the formation of substitutional C.
71 %The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and concurrently by a reduction of volume due to the lower lattice constant of SiC compared to Si.
72 The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and a concurrent reduction of volume due to the lower lattice constant of SiC compared to Si.
73 Both processes are believed to compensate one another.
75 Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production.
76 It will likewise offer perspectives for processes that rely upon prevention of precipitation events, e.g. the fabrication of strained pseudomorphic Si$_{1-y}$C$_y$ heterostructures\cite{strane96,laveant2002}.
78 Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.
79 In particular, molecular dynamics (MD) constitutes a suitable technique to investigate their dynamical and structural properties.
80 Modelling the processes mentioned above requires the simulation of a large number of atoms ($\approx 10^5-10^6$), which inevitably dictates the atomic interaction to be described by computationally efficient classical potentials.
81 These are, however, less accurate compared to quantum-mechanical methods and their applicability for the description of the physical problem has to be verified beforehand.
82 The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85}, Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic potential\cite{bazant96,bazant97,justo98}.
83 These potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first-principles methods.
84 Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials, e.g. the one by Gao and Weber\cite{gao02} as well as the one by Erhart and Albe\cite{albe_sic_pot}.
85 All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero in between the first and second nearest neighbor distance.
86 In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
87 However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two neighbored atoms.
88 %In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
89 In a previous study\cite{zirkelbach10a} we determined the influence on the migration barrier for C diffusion in Si.
90 Using the Erhart/Albe (EA) potential\cite{albe_sic_pot}, an overestimated barrier height compared to ab initio calculations and experiment is obtained.
91 A proper description of C diffusion, however, is crucial for the problem under study.
93 In this work, a combined ab initio and empirical potential simulation study on the initially mentioned SiC precipitation mechanism has been performed.
94 Highly accurate quantum-mechanical results have been used to identify shortcomings of the classical potentials, which are then taken into account in these type of simulations.
96 % --------------------------------------------------------------------------------
99 The first-principles DFT calculations have been performed with the plane-wave based Vienna ab initio Simulation package (VASP)\cite{kresse96}.
100 The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
101 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
102 Throughout this work, an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
103 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
104 The defect structures and the migration paths have been modeled in cubic supercells containing 216 Si atoms.
105 The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
106 Spin polarization has been fully accounted for.
108 % ------ Albe potential ---------
109 For the classical potential calculations, defect structures were modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms.
110 Reproducing the SiC precipitation was attempted by the successive insertion of 6000 C atoms (the number necessary to form a 3C-SiC precipitate with a radius of $\approx 3.1$ nm) into the Si host, which has a size of 31 Si unit cells in each direction consisting of 238328 Si atoms.
111 At constant temperature 10 atoms were inserted at a time.
112 Three different regions within the total simulation volume were considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, which holds the necessary amount of Si atoms of the precipitate.
113 After C insertion the simulation has been continued for \unit[100]{ps} and is cooled down to \unit[20]{$^{\circ}$C} afterwards.
114 A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions realized by a cut-off function dropping the interaction to zero in between the first and second nearest neighbor distance.
115 The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
116 Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
117 The temperature was kept constant by the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs}.
118 Integration of the equations of motion was realized by the velocity Verlet algorithm\cite{verlet67} and a fixed time step of \unit[1]{fs}.
119 For structural relaxation of defect structures the same algorithm was used with the temperature set to 0 K.
121 The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ of a defect configuration is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
122 Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique\cite{kaukonen98}.
123 Time constants of \unit[1]{fs}, which corresponds to direct velocity scaling, and \unit[100]{fs}, which results in weaker coupling to the heat bath allowing the diffusing atoms to take different pathways, were used for the Berendsen thermostat for structural relaxation within the migration calculations.
127 \subsection{Carbon and silicon defect configurations}
129 Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations.
132 \begin{tabular}{l c c c c c c}
133 & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ T\\
135 VASP & 3.72 & 1.95 & 4.16 & 4.66 & 3.39 & 3.77 \\
136 Erhart/Albe & 3.88 & 0.75 & 5.18 & 5.59$^*$ & 4.39 & 3.40
139 \caption{Formation energies of C and Si point defects in c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volts. T denotes the tetrahedral and BC the bond-centered configuration. Subscript i and s indicates the interstitial and substitutional configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
142 Although discrepancies exist, both methods depict the correct order of the formation energies with regard to C defects in Si.
143 Substitutional C (C$_{\text{s}}$) constitutes the energetically most favorable defect configuration.
144 Since the C atom occupies an already vacant Si lattice site, C$_{\text{s}}$ is not an interstitial defect.
145 The quantum-mechanical result agrees well with the result of another ab initio study\cite{dal_pino93}.
146 Clearly, the empirical potential underestimates the C$_{\text{s}}$ formation energy.
147 The C interstitial defect with the lowest energy of formation has been found to be the C-Si \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB), which, thus, constitutes the ground state of an additional C impurity in otherwise perfect c-Si.
148 This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
149 Astonishingly EA and DFT predict almost equal formation energies.
150 There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
151 Since the energetic description is considered more important than the structural description, minor discrepancies of the latter are assumed non-problematic.
152 The second most favorable configuration is the C$_{\text{i}}$ \hkl<1 1 0> DB followed by the C$_{\text{i}}$ bond-centered (BC) configuration.
153 For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
154 Thus, nearly the same difference in energy has been observed for these configurations in both methods.
155 However, we have found the BC configuration to constitute a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
156 Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
157 A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations can be found in our recently published article\cite{zirkelbach10a}.
159 Regarding intrinsic defects in Si, both methods predict energies of formation that are within the same order of magnitude.
160 However discrepancies exist.
161 Quantum-mechanical results reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB to compose the energetically most favorable configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
162 The EA potential does not reproduce the correct ground state.
163 Instead the tetrahedral defect configuration is favored.
164 This limitation is assumed to arise due to the cut-off.
165 In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
166 Indeed, an increase of the cut-off results in increased values of the formation energies\cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
167 The same issue has already been discussed by Tersoff\cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
168 While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the following investigations of defect combinations.
169 %This artifact does not necessarily render impossible further challenging empirical potential studies on large systems.
170 %However, it has to be taken into account in the following investigations of defect combinations.
172 \subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
174 As has been shown in a previous study\cite{zirkelbach10b}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
175 However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
177 For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
178 This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
179 A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.
181 Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground state configuration of Si$_{\text{i}}$ in Si it is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.
182 Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
183 Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
184 Results of VASP and EA calculations are summarized in Table~\ref{tab:defect_combos}.
187 \begin{tabular}{l c c c}
188 & C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
190 VASP & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
191 Erhart/Albe & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$
194 \caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volts. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third nearest neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
195 \label{tab:defect_combos}
197 Obviously the EA potential properly describes the relative energies of formation.
198 Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
199 With increasing separation distance the energies of formation decrease.
200 However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
201 Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach10b}, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
202 This is attributed to an effective reduction in strain enabled by the respective combination.
203 Quantum-mechanical results reveal a more favorable energy of fomation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
204 However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> interstitial, thus, not maintaining the tetrahedral Si nor the substitutional C defect.
205 In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
206 Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
208 \subsection{Carbon mobility}
209 \label{subsection:cmob}
211 To accurately model the SiC precipitation, which involves the agglomeration of C, a proper description of the migration process of the C impurity is required.
212 As shown in a previous study\cite{zirkelbach10a}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
213 During transition a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates towards a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
214 However, it turned out that the description fails if the EA potential is used, which overestimates the migration barrier (\unit[2.2]{eV}) by a factor of 2.4.
215 In addition a different diffusion path is found to exhibit the lowest migration barrier.
216 A C$_{\text{i}}$ \hkl[0 0 -1] DB turns into the \hkl[0 0 1] configuration at the neighbored lattice site.
217 The transition involves the C$_{\text{i}}$ BC configuration, which, however, was found to be unstable relaxing into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration.
218 If the migration is considered to occur within a single step, the kinetic energy of \unit[2.2]{eV} is sufficient to turn the \hkl<1 0 0> DB into the BC and back into a \hkl<1 0 0> DB configuration.
219 If, on the other hand, a two step process is assumed, the BC configuration will most probably relax into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration resulting in different relative energies of the intermediate state and the saddle point.
220 For the latter case a migration path, which involves a C$_{\text{i}}$ \hkl<1 1 0> DB configuration, is proposed and displayed in Fig.~\ref{fig:mig}.
223 \includegraphics[width=\columnwidth]{110mig.ps}
225 \caption{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations were performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
228 Approximately \unit[2.24]{eV} are needed to turn the C$_{\text{i}}$ \hkl[0 0 -1] DB into the C$_{\text{i}}$ \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
229 Another barrier of \unit[0.90]{eV} exists for the rotation into the C$_{\text{i}}$ \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
230 Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in our previous study\cite{zirkelbach10a}.
231 The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
232 By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by ab initio calculations.
234 Accordingly, the effective barrier of migration of C$_{\text{i}}$ is overestimated by a factor of 2.4 to 3.5 compared to the highly accurate quantum-mechanical methods.
235 This constitutes a serious limitation that has to be taken into account for modeling the C-Si system using the otherwise quite promising EA potential.
237 \subsection{Molecular dynamics simulations}
239 Fig.~\ref{fig:450} shows the radial distribution functions of simulations, in which C was inserted at \unit[450]{$^{\circ}$C}, an operative and efficient temperature in IBS\cite{lindner99}, for all three insertion volumes.
242 \subfigure[]{\label{fig:450:a}
243 \includegraphics[width=\columnwidth]{../img/sic_prec_450_si-si_c-c.ps}
245 \subfigure[]{\label{fig:450:b}
246 \includegraphics[width=\columnwidth]{../img/sic_prec_450_si-c.ps}
249 \caption{Radial distribution function for C-C and Si-Si (Fig.~\ref{fig:450:a}) as well as Si-C (Fig.~\ref{fig:450:b}) pairs for C inserted at \unit[450]{$^{\circ}$C}. In the latter case the resulting C-Si distances for a C$_{\text{i}}$ \hkl<1 0 0> DB are given additionally and the Si-C cut-off distance is marked by an arrow. Insets in Fig.~\ref{fig:450:a} show magnified regions of the respective distribution functions.}
252 There is no significant difference between C insertion into $V_2$ and $V_3$.
253 Thus, in the following, the focus is on low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations only.
255 In the low C concentration simulation the number of C-C bonds is small, as can be seen in the upper part of Fig.~\ref{fig:450:a}.
256 On average, there are only 0.2 C atoms per Si unit cell.
257 By comparing the Si-C peaks of the low concentration simulation with the resulting Si-C distances of a C$_{\text{i}}$ \hkl<1 0 0> DB in Fig.~\ref{fig:450:b} it becomes evident that the structure is clearly dominated by this kind of defect.
258 One exceptional peak at \unit[0.26]{nm} (marked with an arrow in Fig.~\ref{fig:450:b}) exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
259 Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the upper inset of Fig.~\ref{fig:450:a}, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
260 Additionally, in the inset of the bottom part of Fig.\ref{fig:450:a} the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two neighbored Si atoms.
261 This is accompanied by a reduction of the number of bonds at regular Si distances of c-Si.
262 A more detailed description of the resulting C-Si distances in the C$_{\text{i}}$ \hkl<1 0 0> DB configuration and the influence of the defect on the structure is available in a previous study\cite{zirkelbach09}.
264 For high C concentrations, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
265 A subsequent superposition of defects generates new displacement arrangements for the C-C as well as Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
266 Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but hardly visible is the long range order.
267 This indicates the formation of an amorphous SiC-like phase.
268 In fact, 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}.
270 In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
271 With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
272 However, sufficient defect agglomeration is not observed.
273 For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
274 On closer inspection two reasons for describing this obstacle become evident.
276 First of all, there is the time scale problem inherent to MD in general.
277 To minimize the integration error the discretized time step must be chosen smaller than the reciprocal of the fastest vibrational mode resulting in a time step of \unit[1]{fs} for the investigated materials system.
278 Limitations in computer power result in a slow propagation in phase space.
279 Several local minima exist, which are separated by large energy barriers.
280 Due to the low probability of escaping such a local minimum, a single transition event corresponds to a multiple of vibrational periods.
281 Long-term evolution, such as a phase transformation and defect diffusion, in turn, are made up of a multiple of these infrequent transition events.
282 Thus, time scales to observe long-term evolution are not accessible by traditional MD.
283 New accelerated methods have been developed to bypass the time scale problem retaining proper thermodynamic sampling\cite{voter97,voter97_2,voter98,sorensen2000,wu99}.
285 However, the applied potential comes up with an additional limitation, as previously mentioned in the introduction.
286 The cut-off function of the short range potential limits the interaction to nearest neighbors, which results in overestimated and unphysical high forces between neighbored atoms.
287 This behavior, as observed and discussed for the Tersoff potential\cite{tang95,mattoni2007}, is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:cmob}.
288 Indeed, it is not only the strong, hard to break C-C bond inhibiting C diffusion and further rearrangements in the case of the high C concentration simulations.
289 This is also true for the low concentration simulations dominated by the occurrence of C$_{\text{i}}$ \hkl<1 0 0> DBs spread over the whole simulation volume, which are unable to agglomerate due to the high migration barrier.
291 \subsection{Increased temperature simulations}
293 Due to the problem of slow phase space propagation, which is enhanced by the employed potential, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively might not be sufficient.
294 Instead, higher temperatures are utilized to compensate overestimated diffusion barriers.
295 These are overestimated by a factor of 2.4 to 3.5.
296 Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460-2260]{$^{\circ}$C}.
297 Since melting already occurs shortly below the melting point of the potential (2450 K)\cite{albe_sic_pot} due to the presence of defects, a maximum temperature of \unit[2050]{$^{\circ}$C} is used.
299 Fig.~\ref{fig:tot} shows the resulting radial distribution functions for various temperatures.
302 \subfigure[]{\label{fig:tot:si-c}
303 \includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}
305 \subfigure[]{\label{fig:tot:si-si}
306 \includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}
308 \subfigure[]{\label{fig:tot:c-c}
309 \includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
312 \caption{Radial distribution function for Si-C (Fig.~\ref{fig:tot:si-c}), Si-Si (Fig.~\ref{fig:tot:si-si}) and C-C (Fig.~\ref{fig:tot:c-c}) pairs for the C insertion into $V_1$ at elevated temperatures. For the Si-C distribution resulting Si-C distances of a C$_{\text{s}}$ configuration are plotted. In the C-C distribution dashed arrows mark C-C distances occurring from C$_{\text{i}}$ \hkl<1 0 0> DB combinations, solid arrows mark C-C distances of pure C$_{\text{s}}$ combinations and the dashed line marks C-C distances of a C$_{\text{i}}$ and C$_{\text{s}}$ combination.}
315 In Fig.~\ref{fig:tot:si-c}, the first noticeable and promising change observed for the Si-C bonds is the successive decline of the artificial peak at the cut-off distance with increasing temperature.
316 Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
317 Additionally, a more important structural change was observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:tot:si-c}.
318 Obviously, the structure obtained at \unit[450]{$^{\circ}$C}, which was found to be dominated by C$_{\text{i}}$, transforms into a C$_{\text{s}}$ dominated structure with increasing temperature.
319 Comparing the radial distribution at \unit[2050]{$^{\circ}$C} to the resulting bonds of C$_{\text{s}}$ in c-Si excludes all possibility of doubt.
321 The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm} in Fig.~\ref{fig:tot:si-si}, which corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
322 Since the expected distance of these Si pairs in 3C-SiC is \unit[0.308]{nm} the existing SiC structures embedded in the c-Si host are stretched.
324 According to the C-C radial distribution displayed in Fig.~\ref{fig:tot:c-c}, agglomeration of C fails to appear even for elevated temperatures, as can be seen on the total amount of C pairs within the investigated separation range, which does not change significantly.
325 However, a small decrease in the amount of neighbored C pairs can be observed with increasing temperature.
326 This high temperature behavior is promising since breaking of these diamond- and graphite-like bonds is mandatory for the formation of 3C-SiC.
327 Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
328 A slight shift towards higher distances can be observed for the maximum located shortly above \unit[0.3]{nm}.
329 Arrows with dashed lines mark C-C distances resulting from C$_{\text{i}}$ \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of C$_{\text{s}}$.
330 The continuous dashed line corresponds to the distance of C$_{\text{s}}$ and a neighbored C$_{\text{i}}$ DB.
331 Obviously, the shift of the peak is caused by the advancing transformation of the C$_{\text{i}}$ DB into the C$_{\text{s}}$ defect.
332 Quite high g(r) values are obtained for distances in between the continuous dashed line and the first arrow with a solid line.
333 For the most part, these structures can be identified as configurations of C$_{\text{s}}$ with either another C atom that basically occupies a Si lattice site but is displaced by a Si interstitial residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
334 Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
336 Fig.~\ref{fig:v2} displays the radial distribution for high C concentrations.
339 \subfigure[]{\label{fig:v2:si-c}
340 \includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}
342 \subfigure[]{\label{fig:v2:c-c}
343 \includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
346 \caption{Radial distribution function for Si-C (Fig.~\ref{fig:v2:si-c}) and C-C (Fig.~\ref{fig:v2:c-c}) pairs for the C insertion into $V_2$ at elevated temperatures. Arrows mark the respective cut-off distances.}
351 \includegraphics[width=\columnwidth]{../img/2050.eps}
353 \caption{Cross section along the \hkl(1 -1 0) plane of the atomic structure of the high concentration simulation for a C insertion temperature of \unit[2050]{$^{\circ}$C}.}
356 A cross-section along the \hkl(1 -1 0) plane of the atomic structure for a C insertion temperature of \unit[2050]{$^{\circ}$C} is shown in Fig.~\ref{fig:v2as}.
357 The amorphous SiC-like phase remains.
358 No significant change in structure is observed.
359 However, the decrease of the cut-off artifact and slightly sharper peaks observed with increasing temperature, in turn, indicate a slight acceleration of the dynamics realized by the supply of kinetic energy.
360 However, it is not sufficient to enable the amorphous to crystalline transition.
361 In contrast, even though bonds of neighbored C atoms could be partially dissolved in the system exhibiting low C concentrations, the amount of neighbored C pairs even increased in the latter case.
362 Moreover, the C-C peak at \unit[0.252]{nm} in Fig.~\ref{fig:v2:c-c}, which gets slightly more distinct, equals the second nearest neighbor distance in diamond and indeed is made up by a structure of two C atoms interconnected by a third C atom.
363 Obviously, processes that appear to be non-conducive are likewise accelerated in a system, in which high amounts of C are incorporated within a short period of time, which is accompanied by a concurrent introduction of accumulating, for the reason of time non-degradable damage.
364 % non-degradable, non-regenerative, non-recoverable
365 Thus, for these systems even larger time scales, which are not accessible within traditional MD, must be assumed for an amorphous to crystalline transition or structural evolution in general.
366 % maybe put description of bonds in here ...
367 Nevertheless, some results likewise indicate the acceleration of other processes that, again, involve C$_{\text{s}}$.
368 The increasingly pronounced Si-C peak at \unit[0.35]{nm} in Fig.~\ref{fig:v2:si-c} corresponds to the distance of a C and a Si atom interconnected by another Si atom.
369 Additionally, the C-C peak at \unit[0.31]{nm} in Fig.~\ref{fig:v2:c-c} corresponds to the distance of two C atoms bound to a central Si atom.
370 For both structures the C atom appears to reside on a substitutional rather than an interstitial lattice site.
371 However, huge amounts of damage hamper identification.
372 The alignment of the investigated structures to the c-Si host is lost in many cases, which suggests the necessity of much more time for structural evolution to maintain the topotactic orientation of the precipitate.
374 \section{Discussion and Summary}
376 Investigations are targeted at the initially stated controversy of SiC precipitation, i.e. whether precipitation occurs abruptly after enough C$_{\text{i}}$ agglomerated or after a successive agglomeration of C$_{\text{s}}$ on usual Si lattice sites (and Si$_{\text{i}}$) followed by a contraction into incoherent SiC.
377 Results of a previous ab initio study on defects and defect combinations in C implanted Si\cite{zirkelbach10b} suggest C$_{\text{s}}$ to play a decisive role in the precipitation of SiC in Si.
378 To support previous assumptions MD simulations, which are capable of modeling the necessary amount of atoms, i.e. the precipitate and the surrounding c-Si structure, have been employed in the current study.
380 In a previous comparative study\cite{zirkelbach10a} we have shown that the utilized empirical potential fails to describe some selected processes.
381 Thus, limitations of the employed potential have been further investigated and taken into account in the present study.
382 We focussed on two major shortcomings: the overestimated activation energy and the improper description of intrinsic and C point defects in Si.
383 Overestimated forces between nearest neighbor atoms that are expected for short range potentials\cite{mattoni2007} have been confirmed to influence the C$_{\text{i}}$ diffusion.
384 The migration barrier was estimated to be larger by a factor of 2.4 to 3.5 compared to highly accurate quantum-mechanical calculations\cite{zirkelbach10a}.
385 Concerning point defects, the drastically underestimated formation energy of C$_{\text{s}}$ and deficiency in the description of the Si$_{\text{i}}$ ground state necessitated further investigations on structures that are considered important for the problem under study.
386 It turned out that the EA potential still favors a C$_{\text{i}}$ \hkl<1 0 0> DB over a C$_{\text{s}}$-Si$_{\text{i}}$ configuration, which, thus, does not constitute any limitation for the simulations aiming to resolve the present controversy of the proposed SiC precipitation models.
388 MD simulations at temperatures used in IBS resulted in structures that were dominated by the C$_{\text{i}}$ \hkl<1 0 0> DB and its combinations if C is inserted into the total volume.
389 Incorporation into volumes $V_2$ and $V_3$ led to an amorphous SiC-like structure within the respective volume.
390 To compensate overestimated diffusion barriers, we performed simulations at accordingly increased temperatures.
391 No significant change was observed for high C concentrations.
392 The amorphous phase is maintained.
393 Due to the incorporation of a huge amount of C into a small volume within a short period of time, damage is produced, which obviously decelerates structural evolution.
394 For the low C concentrations, time scales are still too low to observe C agglomeration sufficient for SiC precipitation, which is attributed to the slow phase space propagation inherent to MD in general.
395 However, we observed a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure.
396 The amount of substitutionally occupied C atoms increases with increasing temperature.
397 Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
398 Indeed, in a previous ab initio MD simulation\cite{zirkelbach10b} performed at \unit[900]{$^{\circ}$C} we observed the departing of a Si$_{\text{i}}$ \hkl<1 1 0> DB located next to a C$_{\text{s}}$ atom instead of a recombination into the ground state configuration, i.e. a C$_{\text{i}}$ \hkl<1 0 0> DB.
400 % postannealing less efficient than hot implantation
401 Experimental studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates\cite{eichhorn02}.
402 In particular, restructuring of strong C-C bonds is affected\cite{deguchi92}, which preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements.
403 We assume this to be related to the problem of slow structural evolution encountered in the high C concentration simulations due to the insertion of high amounts of C into a small volume within a short period of time resulting in essentially no time for the system to rearrange.
404 % rt implantation + annealing
405 Implantations of an understoichiometric dose at room temperature followed by thermal annealing results in small spherical sized C$_{\text{i}}$ agglomerates at temperatures below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size at temperatures above \unit[700]{$^{\circ}$C}\cite{werner96}.
406 Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected -- and indeed are observed for as-implanted samples\cite{lindner99,lindner01} -- in implantations performed at \unit[450]{$^{\circ}$C}.
407 Implanted C is therefore expected to occupy substitutionally usual Si lattice sites right from the start.
409 Thus, we propose an increased participation of C$_{\text{s}}$ already in the initial stages of the implantation process at temperatures above \unit[450]{$^{\circ}$C}, the temperature most applicable for the formation of SiC layers of high crystalline quality and topotactical alignment\cite{lindner99}.
410 Thermally activated, C$_{\text{i}}$ is enabled to turn into C$_{\text{s}}$ accompanied by Si$_{\text{i}}$.
411 The associated emission of Si$_{\text{i}}$ is needed for several reasons.
412 For the agglomeration and rearrangement of C, Si$_{\text{i}}$ is needed to turn C$_{\text{s}}$ into highly mobile C$_{\text{i}}$ again.
413 Since the conversion of a coherent SiC structure, i.e. C$_{\text{s}}$ occupying the Si lattice sites of one of the two fcc lattices that build up the c-Si diamond lattice, into incoherent SiC is accompanied by a reduction in volume, large amounts of strain are assumed to reside in the coherent as well as at the surface of the incoherent structure.
414 Si$_{\text{i}}$ serves either as a supply of Si atoms needed in the surrounding of the contracted precipitates or as an interstitial defect minimizing the emerging strain energy of a coherent precipitate.
415 The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms and Si$_{\text{i}}$ located in the vicinity.
417 It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.\cite{nejim95}.
418 This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach10b}, which showed C$_{\text{s}}$ to occur in all probability.
419 However, agglomeration and rearrangement is enabled by mobile C$_{\text{i}}$, which has to be present at the same time and is formed by recombination of C$_{\text{s}}$ and Si$_{\text{i}}$.
420 In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
421 This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
423 % ----------------------------------------------------
424 \section*{Acknowledgment}
425 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).
426 Meta Schnell is greatly acknowledged for a critical revision of the present manuscript.
428 % --------------------------------- references -------------------
430 \bibliography{../../bibdb/bibdb}{}
431 \bibliographystyle{h-physrev3}
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