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 silcon}
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 mapped to feasible systems of small size.
31 Results of the accurate first-principles calculations on the carbon diffusion in silicon are compared to results of calssical 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, ... more ...}
37 \pacs{61.72.uf,66.30.-h,31.15.A-,34.20.Cf, ... more ...}
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 Howeve, the frequent occurrence of defects such as twins, dislocations and double position boundaries remains a challenging problem.
51 Next to 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.
55 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.
56 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 due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
57 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.
58 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.
59 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.
60 These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
61 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.
62 However, in another IBS study Nejim et~al.\cite{nejim95} propose a topotactic transformation that is likewise based on the formation of substitutional C.
63 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.
64 Both processes are believed to compensate each other.
66 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.
67 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}.
69 Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.
70 In particular, molecular dynamics (MD) constitutes a suitable technique to investigate the dynamical and structural properties of some material.
71 Modelling the processes mentioned above requires the simulation of a large amount of atoms ($\approx 10^5-10^6$), which inevitably dictates the atomic interaction to be described by computationally efficient classical potentials.
72 These are, however, less accurate compared to quantum-mechnical methods and theire applicability for the description of the physical problem has to be verified first.
73 The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic potential (EDIP)\cite{bazant96,bazant97,justo98}.
74 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.
75 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}.
76 All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero inbetween the first and second next neighbor distance.
77 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}.
78 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 next neighbored atoms.
79 In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
80 Using the Erhart/Albe (EA) potential\cite{albe_sic_pot} an overestimated barrier height compared to ab initio calculations and experiment is obtained.
81 A proper edscription of C diffusion, however, is crucial for the problem under study.
83 In this work, a combined ab initio and empirical potential simulation study on the initially mentioned SiC precipitation mechanism has been performed.
84 High 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.
86 % --------------------------------------------------------------------------------
89 The first-principles DFT calculations have been performed with the plane-wave based Vienna ab initio Simulation package (VASP)\cite{kresse96}.
90 The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
91 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
92 Throughout this work, an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
93 Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
94 The defect structures and the migration paths have been modeled in cubic supercells containing 216 Si atoms.
95 The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
96 Spin polarization has been fully accounted for.
98 % ------ Albe potential ---------
99 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.
100 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.
101 At constant temperature 10 atoms were inserted at a time.
102 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 $V_3$, whih holds the necessary amount of Si atoms of the precipitate.
103 After C insertion the simulation has been continued for \unit[100]{ps} and cooled down to \unit[20]{$^{\circ}$C} afterwards.
104 A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
105 The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
106 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.
107 The temperature is kept constant by the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs}.
108 Integration of equations of motion is realized by the velocity Verlet algorithm\cite{verlet67} and a fixed time step of \unit[1]{fs}.
109 For structural relaxation of defect structures the same algorith is used with the temperature set to 0 K.
111 The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ of a defect configuration is defined by chosing 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.
112 Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
113 The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.
114 Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.
118 \subsection{Carbon and silicon defect configurations}
120 Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations.
123 \begin{tabular}{l c c c c c c}
124 & 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\\
126 VASP & 3.72 & 1.95 & 4.16 & 4.66 & 3.39 & 3.77 \\
127 Erhart/Albe & 3.88 & 0.75 & 5.18 & 5.59$^*$ & 4.39 & 3.40
130 \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 volt. 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.}
133 Although discrepancies exist, both methods depict the correct order of the formation energies with regard to C defects in Si.
134 Substitutional C (C$_{\text{s}}$) constitutes the energetically most favorable defect configuration.
135 Since the C atom occupies an already vacant Si lattice site, C$_{\text{s}}$ is not an interstitial defect.
136 The quantum-mechanical result agrees well with the result of another ab initio study\cite{dal_pino93}.
137 Clearly, the empirical potential underestimates the C$_{\text{s}}$ formation energy.
138 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.
139 This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
140 Astonishingly EA and DFT predict almost equal formation energies.
141 There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four next neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
142 Since the energetic description is considered more important than the structural description minor discrepancies of the latter are assumed non-problematic.
143 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.
144 For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
145 Thus, nearly the same difference in energy has been observed for these configurations in both methods.
146 However, we have found the BC configuration to constitute a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
147 Due to the high formation energy of the BC defect resulting in a low probability of occurence of this defect, the wrong description is not posing a serious limitation of the EA potential.
148 A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations are presented in a previous study\cite{zirkelbach10a}.
150 Regarding intrinsic defects in Si, both methods predict energies of formation that are within the same order of magnitude.
151 However discrepancies exist.
152 Quantum-mechanical results reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB to compose the energetically most favorabe configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
153 The EA potential does not reproduce the correct ground state.
154 Instead the tetrahedral defect configuration is favored.
155 This limitation is assumed to arise due to the cut-off.
156 In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
157 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.
158 The same issue has already been discussed by Tersoff\cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
159 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.
160 %This artifact does not necessarily render impossible further challenging empirical potential studies on large systems.
161 %However, it has to be taken into account in the following investigations of defect combinations.
163 \subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
165 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 next neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
166 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.
168 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$_{\text{i}}$.
169 This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
170 A possible occurence 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 obviously is wrong.
172 Since quantum-mechanical calculation 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$_{\text{s}}$.
173 Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
174 Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
175 Results of VASP and EA calculations are summarized in Table~\ref{tab:defect_combos}.
178 \begin{tabular}{l c c c}
179 & C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
181 VASP & 3.72 & 4.37 & - \\
182 Erhart/Albe & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$
185 \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 volt. 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 next neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
186 \label{tab:defect_combos}
188 Obviously the EA potential properly describes the relative energies of formation.
189 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.
190 With increasing separation distance the enrgies of formation decrease.
191 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.
192 Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a next neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical caluclations\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.
193 This is attributed to an effective reduction in strain enabled by the respective combination.
194 Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
196 \subsection{Carbon mobility}
197 \label{subsection:cmob}
199 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.
200 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}.
201 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 next neighbored lattice site in \hkl[1 1 -1] direction.
202 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.
203 In addition a different diffusion path is found to exhibit the lowest migration barrier.
204 A C$_{\text{i}}$ \hkl[0 0 -1] DB turns into the \hkl[0 0 1] configuration at the next neighbored lattice site.
205 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.
206 If the migration is considered to occur within a single step the kinetic energy of \unit[2.2]{eV} is enough to turn the \hkl<1 0 0> DB into the BC and back into a \hkl<1 0 0> DB configuration.
207 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.
208 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}.
211 \includegraphics[width=\columnwidth]{110mig.ps}
213 \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.}
216 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 next neighbored lattice site in \hkl[1 1 -1] direction.
217 Another barrier of \unit[0.90]{eV} exists for the rotation into the C$_{\text{i}}$ \hkl[0 -1 0] DB configuration.
218 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}.
219 The former diffusion process, however, would more nicely agree to the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
220 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.
222 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.
223 This constitutes a serious limitation that has to be taken into account for modeling the C-Si system using the EA potential.
225 \subsection{Molecular dynamics simulations}
227 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, for all three insertion volumes.
230 \includegraphics[width=\columnwidth]{../img/sic_prec_450_si-si_c-c.ps}\\
231 \includegraphics[width=\columnwidth]{../img/sic_prec_450_si-c.ps}
233 \caption{Radial distribution function for C-C and Si-Si (top) as well as Si-C (bottom) 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.}
236 There is no significant difference between C insertion into $V_2$ and $V_3$.
237 Thus, in the following, the focus is on low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations only.
239 In the low C concentration simulation the number of C-C bonds is small.
240 On average, there are only 0.2 C atoms per Si unit cell.
241 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 it becomes evident that the structure is clearly dominated by this kind of defect.
242 One exceptional peak exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
243 Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the inset, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
244 Additionally 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 next neighbored Si atoms.
245 This is accompanied by a reduction of the number of bonds at regular Si distances of c-Si.
246 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}.
248 For high C concentrations the defect concentration is likewise increased and a considerable amount of damamge is introduced in the insertion volume.
249 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.
250 Short range order indeed is observed, i.e. the large amount of strong next 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 only hardly visible is the long range order.
251 This indicates the formation of an amorphous SiC-like phase.
252 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 modifed Tersoff potential\cite{gao02}.
254 In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
255 With respect to the precipitation model the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
256 However, sufficient defect agglomeration is not observed.
257 For high C concentrations a rearrangment of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
258 On closer inspection two reasons for describing this obstacle become evident.
260 First of all there is the time scale problem inherent to MD in general.
261 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 current problem under study.
262 Limitations in computer power result in a slow propgation in phase space.
263 Several local minima exist, which are separated by large energy barriers.
264 Due to the low probability of escaping such a local minimum a single transition event corresponds to a multiple of vibrational periods.
265 Long-term evolution such as a phase transformation and defect diffusion, in turn, are made up of a multiple of these infrequent transition events.
266 Thus, time scales to observe long-term evolution are not accessible by traditional MD.
267 New accelerated methods have been developed to bypass the time scale problem retaining proper thermodynamic sampling\cite{voter97,voter97_2,voter98,sorensen2000,wu99}.
269 However, the applied potential comes up with an additional limitation already mentioned in the introductory part.
270 The cut-off function of the short range potential limits the interaction to next neighbors, which results in overestimated and unphysical high forces between next neighbor atoms.
271 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}.
272 Indeed it is not only the strong C-C bond which is hard to break inhibiting C diffusion and further rearrengements in the case of the high C concentration simulations.
273 This is also true for the low concentration simulations dominated by the occurrence of C$_{\text{i}}$ \hkl<1 1 0> DBs spread over the whole simulation volume, which are unable to agglomerate due to the high migration barrier.
275 \subsection{Increased temperature simulations}
277 Due to the potential enhanced problem of slow phase space propagation, pushing the time scale to the limits of computational ressources or applying one of the above mentioned accelerated dynamics methods exclusively might not be sufficient.
278 Instead higher temperatures are utilized to compensate overestimated diffusion barriers.
279 These are overestimated by a factor of 2.4 to 3.5.
280 Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460-2260]{$^{\circ}$C}.
281 Since melting already occurs shortly below the melting point of the potetnial (2450 K) due to the defects, a maximum temperature of \unit[2050]{$^{\circ}$C} is used.
282 Fig.~\ref{fig:tot} shows the resulting bonds for various temperatures.
285 \includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}\\
286 \includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}\\
287 \includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
289 \caption{Radial distribution function for Si-C (top), Si-Si (center) and C-C (bottom) pairs for the C insertion into $V_1$ at elevated temperatures. In the latter case dashed arrows mark C-C distances occuring 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.}
292 Obviously a phase transition occurs ... WEITER
297 \includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}\\
298 \includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
300 \caption{Radial distribution function for Si-C (top) and C-C (bottom) pairs for the C insertion into $V_2$ at elevated temperatures.}
306 The first-principles results are in good agreement to previous work on this subject\cite{burnard93,leary97,dal_pino93,capaz94}.
307 The C-Si \hkl<1 0 0> dumbbell interstitial is found to be the ground state configuration of a C defect in Si.
308 The lowest migration path already proposed by Capaz et~al.\cite{capaz94} is reinforced by an additional improvement of the quantitative conformance of the barrier height calculated in this work (\unit[0.9]{eV}) with experimentally observed values (\unit[0.70]{eV} -- \unit[0.87]{eV})\cite{lindner06,song90,tipping87}.
309 However, it turns out that the bond-centered configuration is not a saddle point configuration as proposed by Capaz et~al.\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
310 A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the sp hybridized C atom, is settled.
311 By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
312 With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.
314 We found that the description of the same processes fails if classical potential methods are used.
315 Already the geometry of the most stable dumbbell configuration differs considerably from that obtained by first-principles calculations.
316 The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
317 %ref mod: language - energy / order
318 %Nevertheless, both methods predict the same type of interstitial as the ground state configuration, and also the order in energy of the remaining defects is reproduced fairly well.
319 Nevertheless, both methods predict the same type of interstitial as the ground state configuration.
320 Furthermore, the relative energies of the other defects are reproduced fairly well.
321 From this, a description of defect structures by classical potentials looks promising.
322 % ref mod: language - changed
323 %However, focussing on the description of diffusion processes the situation is changing completely.
324 However, focussing on the description of diffusion processes the situation has changed completely.
325 Qualitative and quantitative differences exist.
326 First of all, a different pathway is suggested as the lowest energy path, which again might be attributed to the absence of quantum-mechanical effects in the classical interaction model.
327 Secondly, the activation energy is overestimated by a factor of 2.4 compared to the more accurate quantum-mechanical methods and experimental findings.
328 This is attributed to the sharp cut-off of the short range potential.
329 As already pointed out in a previous study\cite{mattoni2007} the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
330 The overestimated migration barrier, however, affects the diffusion behavior of the C interstitials.
331 By this artifact the mobility of the C atoms is tremendously decreased resulting in an inaccurate description or even absence of the dumbbell agglomeration as proposed by the precipitation model.
335 To conclude, we have shown that ab initio calculations on interstitial carbon in silicon are very close to the results expected from experimental data.
336 The calculations presented in this work agree well with other theoretical results.
337 So far, the best quantitative agreement with experimental findings has been achieved concerning the interstitial carbon mobility.
338 For the first time, we have shown that the bond-centered configuration indeed constitutes a real local minimum configuration resulting in a net magnetization if spin polarized calculations are performed.
339 Classical potentials, however, fail to describe the selected processes.
340 This has been shown to have two reasons, i.e. the overestimated barrier of migration due to the artificial interaction cut-off on the one hand, and on the other hand the lack of quantum-mechanical effects which are crucial in the problem under study.
341 % ref mod: language - being investigated
342 %In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
343 In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently being performed.
345 % ----------------------------------------------------
346 \section*{Acknowledgment}
347 We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).
349 % --------------------------------- references -------------------
351 \bibliography{../../bibdb/bibdb}{}
352 \bibliographystyle{h-physrev3}
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