1 \chapter{Summary and conclusions}
2 \label{chapter:summary}
5 in a short review of the C/Si compound and the fabrication of the technologically promising semiconductor SiC by IBS, two controversial assumptions of the precipitation mechanism of 3C-SiC in c-Si are elaborated.
6 These propose precipitation of SiC by agglomeration of \ci{} DBs followed by a sudden formation of SiC and otherwise a formation by successive accumulation of \cs{} via intermediate stretched SiC structures, which are coherent to the Si lattice.
7 To solve this controversy and contribute to the understanding of SiC precipitation in c-Si, a series of atomistic simulations is carried out.
8 In the first part, intrinsic and C related point defects in c-Si as well as some selected diffusion processes of the C defect are investigated by means of first-principles quantum-mechanical calculations based on DFT and classical potential calculations employing a Tersoff-like analytical bond order potential.
9 Shortcomings of the computationally efficient though less accurate classical potential approach compared to the quantum-mechanical treatment are revealed.
10 The study proceeds investigating combinations of defect structures and related diffusion processes exclusively by the first-principles method.
11 The applicability of the utilized bond order potential for subsequent MD simulations is discussed.
12 Conclusions on the precipitation mechanism based on the DFT results are drawn.
13 In the second part, classical potential MD simulations are performed, which try to directly reproduce the precipitation process.
14 Next to the shortcomings of the potential, quirks inherent to MD are discussed and a workaround is proposed.
15 Although direct formation of SiC fails to appear, the obtained results indicate a mechanism of precipitation, which is consistent with previous quantum-mechanical conclusions as well as experimental findings.
17 Quantum-mechanical results of intrinsic point defects in Si are in good agreement to previous theoretical work on this subject~\cite{leung99,al-mushadani03}.
18 The \si{} \hkl<1 1 0> DB defect is reproduced as the ground-state configuration followed by the hexagonal and tetrahedral defect.
19 Spin polarized calculations are required for the \si{} \hkl<1 0 0> DB and vacancy whereas no other of the investigated intrinsic defects is affected.
20 For the \si{} \hkl<1 0 0> DB, the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms.
21 For the vacancy, the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
22 Results obtained by calculations utilizing the classical EA potential yield formation energies, which are of the same order of magnitude.
23 However, EA predicts the tetrahedral configuration to be most stable.
24 The particular problem is due to the cut-off and the fact that the second neighbors are only slightly more distant than the first neighbors within the tetrahedral configuration.
25 Furthermore, the hexagonal defect structure is not stable opposed to results of the authors of the potential~\cite{albe_sic_pot}.
26 The obtained structure after relaxation, which is similar to the tetrahedral configuration, exhibits a formation energy equal to the one given by the authors for the hexagonal one.
27 Obviously, the authors did not check the relaxed structure still assuming a hexagonal configuration.
28 The actual structure is equal to the tetrahedral configuration, which is slightly displaced along the three coordinate axes.
29 Variations exist with displacements along two or a single \hkl<1 0 0> direction indicating a potential artifact.
30 However, finite temperature simulations are not affected by this artifact due to a low activation energy necessary for a transition into the energetically more favorable tetrahedral configuration.
31 Next to the known problem of the underestimated formation energy of the tetrahedral configuration~\cite{tersoff90}, the energetic sequence of the defect structures is well reproduced by the EA calculations.
32 Migration barriers of \si{} investigated by quantum-mechanical calculations are found to be of the same order of magnitude than values derived in other {\em ab initio} studies~\cite{bloechl93,sahli05}.
34 Defects of C in Si are well described by both methods.
35 The \ci{} \hkl<1 0 0> DB is found to constitute the most favorable interstitial configuration in agreement with several theoretical~\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental~\cite{watkins76,song90} investigations.
36 Almost equal formation energies are predicted by the EA and DFT calculations for this defect.
37 A small discrepancy in the resulting equilibrium structure with respect to the DFT method exists due to missing quantum-mechanical effects within the classical treatment.
38 The high formation energies of the tetrahedral and hexagonal configuration obtained by classical potentials act in concert with the fact that these configurations are found unstable by the first-principles description.
39 The BC configuration turns out to be unstable relaxing into the \ci{} \hkl<1 1 0> DB configuration within the EA approach.
40 This is supported by another {\em ab initio} study~\cite{capaz94}, which in turn finds the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
41 By quantum-mechanical calculations performed in this work, however, it turns out that the BC configuration constitutes a real local minimum if the electron spin is fully accounted for.
42 Indeed, spin polarization is absolutely necessary for the BC configuration resulting in a net magnetization of two electrons accompanied by a reduction of the total energy by \unit[0.3]{eV}.
43 The resulting spin up density is localized in a torus around the C perpendicular to the linear Si-C-Si bond.
44 No other configuration is affected by spin polarization.
45 The underestimated formation energy of \cs{} is a definite drawback of the classical potential.
46 However, the creation of \cs{} is necessarily accompanied by a \si{} in a perfect Si crystal, in which a C atom is incorporated.
47 Fortunately, the energetics of combinations of \cs{} and \si{} are quite well described by the EA potential.
48 Thus, the underestimated formation energy does not pose a serious limitation.
49 Based on the above findings, it is concluded that modeling of the SiC precipitation by the EA potential might lead to trustable results.
51 Quantum-mechanical investigations of the mobility of the \ci{} \hkl<1 0 0> DB yield a migration barrier of \unit[0.9]{eV}, which excellently agrees to experimental values ranging from \unit[0.70]{eV} to \unit[0.87]{eV}~\cite{lindner06,song90,tipping87}.
52 The respective path corresponds to a \ci{} \hkl[0 0 -1] DB migrating towards the next neighbored Si atom located in \hkl[1 1 -1] direction forming a \ci{} \hkl[0 -1 0] DB.
53 The identified migration path involves a change in orientation of the DB.
54 Thus, the same path explains the experimentally determined activation energies for reorientation of the DB ranging from \unit[0.77]{eV}~\cite{watkins76} up to \unit[0.88]{eV}~\cite{song90}.
55 Investigations based on the EA bond order potential suggest a migration involving an intermediate \ci{} \hkl<1 1 0> DB configuration.
56 Although different, starting and final configuration as well as the change in orientation of the \hkl<1 0 0> DB are equal to the identified pathway by the {\em ab initio} calculations.
57 However, barrier heights, which are overestimated by a factor of 2.4 to 3.5 depending on the character of migration, i.e.\ a single step or two step process, compared to the DFT results, are obtained.
58 Obviously, the EA potential fails to describe \ci{} diffusion yielding a drastically overestimated activation energy, which has to be taken into account in subsequent investigations.
60 A series of investigations focuses on defect combinations exclusively by the first-principles description.
61 These configurations are constructed in such a way as to allow for a quantum-mechanical treatment.
63 Investigations of two \ci{} defects of the \hkl<1 0 0>-type for varying separations and orientations state a fairly attractive interaction between these interstitials.
64 The capture radius is found to be rather large compared to other defect combinations.
65 For the most part, energetically favorable configurations of two interstitials are found.
66 This is due to strain compensation enabled by the combination of such defects in certain orientations.
67 An interaction energy proportional to the reciprocal cube of the distance in the far field regime is found supporting the assumption of \ci{} DB agglomeration.
68 The energetically most favorable configuration consists of a C-C bond.
69 However, due to high activation energies of respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations, this structure is less likely to arise than structures of C atoms that are interconnected by another Si atom.
70 Thus, agglomeration of C$_{\text{i}}$ is expected whereas the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.
72 Results of combinations of \ci{} and \cs{} revealed two additional metastable structures different to these obtained by a naive relaxation.
73 Small displacements result in a structure of a \hkl<1 1 0> C-C DB and in a structure of a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
74 Both structures are lower in energy compared to the respective counterparts.
75 These results, for the most part, compare well with results gained in previous studies~\cite{leary97,capaz98,liu02} and show an astonishingly good agreement with experiment~\cite{song90_2}.
76 Again, spin polarized calculations are revealed necessary.
77 A net magnetization of two electrons is observed for the \hkl<1 1 0> C-C DB configuration, which constitutes the ground state.
78 A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0] due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom.
79 All other investigated configurations show attractive interactions, which suggest an energetically favorable agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ except for separations along one of the \hkl<1 1 0> directions.
80 Although the most favorable configuration exhibits a C-C bond, migration paths show large barriers exceeding \unit[2.2]{eV} for transitions into the ground state.
81 As before, structures other than the ground-state configuration are assumed to arise more likely.
82 Thus, agglomeration of C defects in contrast to C clustering is again reinforced by these findings.
84 C$_{\text{i}}$ and vacancies are found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in stable C$_{\text{s}}$ configurations.
85 In addition, a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects, is observed.
86 Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.
87 Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
88 Thus, C interstitials and vacancies located close together are assumed to end up in a configuration of \cs{}.
90 Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
91 However, a small capture radius is identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground-state configuration.
92 In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.
93 Low diffusion barriers of \si{} enable further separation of the defect pair.
94 Thus, elevated temperatures might lead to configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is likewise supported by the result of the MD run.
96 % maybe preliminary conclusions here ...
98 Classical potential MD calculations targeting the direct simulation of SiC precipitation in Si are adopted.
99 Therefor, the necessary amount of C is gradually incorporated into a large c-Si host.
100 Simulations at temperatures used in IBS result in structures dominated by the C$_{\text{i}}$ \hkl<1 0 0> DB and its combinations if C is inserted into the total volume.
101 Incorporation into volumes $V_2$ and $V_3$, which correspond to the volume of the expected precipitate and the volume containing the necessary amount of Si, lead to an amorphous SiC-like structure within the respective volume.
102 Both results are not expected with respect to the outcome of the IBS experiments.
103 In the first case, i.e.\ the low C concentration simulations, \ci{} \hkl<1 0 0> DBs are indeed formed.
104 However, sufficient defect agglomeration is not observed.
105 In the second case, i.e.\ the high C concentration simulations, crystallization of the amorphous structure, which is not expected at prevailing temperatures, is likewise not observed.
107 Limitations of the MD technique in addition to overestimated bond strengths due to the short range potential are identified to be responsible.
108 The approach of using increased temperatures during C insertion is followed to work around this problem termed {\em potential enhanced slow phase space propagation}.
109 Higher temperatures are justified for several reasons.
110 Elevated temperatures are expected to compensate the overestimated diffusion barriers and accelerate structural evolution.
111 In addition, formation of SiC is also observed at higher implantation temperatures~\cite{nejim95,lindner01} and temperatures in the implantation region is definitely higher than the temperature determined experimentally at the surface of the sample.
112 Furthermore, the present study focuses on structural transitions in a system far from equilibrium.
114 No significant change is observed for high C concentrations at increased temperatures.
115 The amorphous phase is maintained.
116 The huge amount of damage hampers identification of investigated structures, which in many cases lost the alignment to the c-Si host.
117 Obviously, incorporation of a high quantity of C into a small volume within a short period of time creates damage, which decelerates structural evolution.
118 For the low C concentrations, time scales are still too low to observe C agglomeration.
119 However, a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure is observed.
120 The amount of \cs{} increases with increasing temperature.
121 Diamond and graphite like bonds as well as the artificial bonds due to the cut-off are reduced.
122 Loose structures of stretched SiC, which are adjusted to the Si lattice with respect to the lattice constant and alignment, are identified.
123 \si{} is often found in the direct surrounding.
124 Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
125 Indeed, utilizing increased temperatures is assumed to constitute a necessary condition to simulate IBS of 3C-SiC in c-Si.
128 % conclusions 2nd part
130 concerning the SiC conversion mechanism are derived from results of both, first-principles and classical potential calculations.
131 Although classical potential MD calculations fail to directly simulate precipitation of SiC, obtained results, on the one hand, reinforce previous findings of the first-principles investigations and, on the other hand, allow further conclusions on the SiC precipitation in Si.
133 Initially, quantum-mechanical investigations suggest agglomeration of \ci{} defects that form energetically favorable configurations by an effective stress compensation.
134 Low barriers of migration are found except for transitions into the ground-state configuration, which is composed of a strong C-C bond.
135 Thus, agglomeration of \ci{} in the absence of C clustering is expected.
136 These initial results suggest a conversion mechanism based on the agglomeration of \ci{} defects followed by a sudden precipitation once a critical size is reached.
137 However, subsequent investigations of structures that are particularly conceivable under conditions prevalent in IBS and at elevated temperatures show \cs{} to occur in all probability.
138 The transition from the ground state of a single C atom incorporated into otherwise perfect c-Si, i.e.\ the \ci{} \hkl<1 0 0> DB, into a configuration of \cs{} next to a \si{} atom exhibits an activation energy lower than the one for the diffusion of the highly mobile \ci{} defect.
139 Considering additionally the likewise lower diffusion barrier of \si{}, configurations of separated \cs{} and \si{} will occur in all probability.
140 This is reinforced by the {\em ab initio} MD run at non-zero temperature, which shows structures of separating instead of recombining \cs{} and \si{} defects.
141 This suggests increased participation of \cs{} already in the initial stages of the implantation process.
142 The highly mobile \si{} is assumed to constitute a vehicle for the rearrangement of other \cs{} atoms onto proper lattice sites, i.e.\ lattice sites of one of the the two fcc lattices composing the diamond structure.
143 This way, stretched SiC structures, which are coherently aligned to the c-Si host, are formed by agglomeration of \cs.
144 Precipitation into an incoherent and partially strain-compensated SiC nucleus occurs once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
145 As already assumed by Nejim~et~al.~\cite{nejim95}, \si{} serves as supply for subsequently implanted C atoms to form further SiC in the resulting free space due to the accompanied volume reduction.
147 Several conclusions based on results obtained by classical potential MD simulations are drawn.
148 First of all, increased temperatures are considered a necessary condition to simulate the IBS of epitaxially aligned 3C-SiC in Si, which constitutes a process far from thermodynamic equilibrium.
149 The strong deviation from equilibrium by elevated temperatures enables the formation of \cs{}-\si{} structures as observed in the quantum-mechanical calculations.
150 In contrast, structures of \ci{} \hkl<1 0 0> DBs, which constitute the thermodynamic ground state, appear at low temperatures.
151 Thus, the mechanism based on the agglomeration of \cs{} is reinforced.
153 Secondly, in configurations of stretched SiC composed by \cs, the accompanied \si{} defect may be assigned further functionality.
154 Next to that as a vehicle that is able to rearrange \cs{} and a building block for the surrounding Si host or further SiC, the analyzed configurations suggest \si{} to be required for stress compensation.
155 As evidently observed in these structures, \si{} reduces tensile strain by capturing a position near one of the C atoms within a configuration of two C atoms that basically reside on Si lattice sites.
156 Furthermore, \si{} might similarly compensate strain in the interface region of an incoherent, nucleated SiC precipitate and the c-Si matrix.
157 %This could be achieved by the formation of \ci{} \hkl<1 0 0> DBs in the Si region slightly contracting the Si atoms next to the C atom to better match the spacing of Si atoms present in 3C-SiC.
158 %Indeed, combinations of \cs{} and \ci{} \hkl<1 0 0> DBs are observed.
160 Further conclusions are derived from results of the high C concentration simulations, in which a large amount of C atoms to obtain stoichiometry is incorporated into a small volume within a short period of time, which results in essentially no time for the system to rearrange.
161 Due to this, the occurrence of strong C-C bonds and the production of a vast amount of damage is observed, which finally results in the formation of an amorphous phase.
162 The strong bonds and damage obviously decelerate structural evolution.
163 The short time, which is not sufficient for structural evolution of the strongly damaged region, can be mapped to a system of low temperature, which lacks the kinetic energy required for the restructuring process.
165 % experimental findings
166 These findings as well as the derived conclusion on the precipitation mechanism involving an increased participation of \cs{} agree well with experimental results.
167 % low t high mobility
168 % high t stable config, no redistr
169 C implanted at room temperature was found to be able to migrate towards the surface in contrast to implantations at \degc{500}, which do not show redistribution of the C atoms~\cite{serre95}.
170 This excellently conforms to the results of the MD simulations at different temperatures, which show the formation of highly mobile \ci{} \hkl<1 0 0> DBs for low and much more stable \cs{} defects for high temperatures.
171 This is likewise suggested by IBS experiments utilizing implantation temperatures of \degc{550} followed by incoherent lamp annealing at temperatures as high as \degc{1405} required for the C segregation due to the stability of \cs{}~\cite{reeson87}.
172 % high imp temps more effective to achieve ?!? ...
173 Furthermore, increased implantation temperatures were found to be more efficient than high temperatures in the postannealing step concerning the formation of topotactically aligned 3C-SiC precipitates~\cite{kimura82,eichhorn02}.
175 Particularly strong C-C bonds, which are hard to break by thermal annealing, were found to effectively aggravate the restructuring process of such configurations~\cite{deguchi92}.
176 These bonds preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements in regions exhibiting a large amount of C atoms.
177 This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations.
179 %Considering the efficiency of high implantation temperatures, experimental arguments exist, which point to the precipitation mechanism based on the agglomeration of \cs.
180 More substantially, understoichiometric implantations at room temperature into preamorphized Si followed by a solid-phase epitaxial regrowth step at \degc{700} result in Si$_{1-x}$C$_x$ layers in the diamond cubic phase with C residing on substitutional Si lattice sites~\cite{strane93}.
181 The strained structure is found to be stable up to \degc{810}.
182 Coherent clustering followed by precipitation is suggested if these structures are annealed at higher temperatures.
184 Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C}~\cite{werner96} annealing temperature.
185 Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples~\cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
186 According to this, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
189 % low t - randomly ...
190 % high t - epitaxial relation ...
191 Moreover, implantations below the optimum temperature for the IBS of SiC show regions of randomly oriented SiC crystallites whereas epitaxial crystallites are found for increased temperatures~\cite{lindner99}.
192 The results of the MD simulations can be interpreted in terms of these experimental findings.
193 The successive occupation of regular Si lattice sites by \cs{} atoms as observed in the high temperature MD simulations and assumed from results of the quantum-mechanical investigations perfectly satisfies the epitaxial relation of substrate and precipitate.
194 In contrast, there is no obvious reason for a topotactic transition of \ci{} \hkl<1 0 0> DB agglomerates, as observed in the low temperature MD simulations, into epitaxially aligned precipitates.
195 The latter transition would necessarily involve a much more profound change in structure.
196 % amorphous region for low temperatures
197 Experimentally, randomly oriented precipitates might also be due to SiC nucleation within the arising amorphous matrix~\cite{lindner99}.
198 In simulation, an amorphous SiC phase is formed for high C concentrations.
199 This is due to high amounts of introduced damage within a short period of time resulting in essentially no time for structural evolution, which is comparable to the low temperature experiments, which lack the kinetic energy necessary for recrystallization of the highly damaged region.
200 Indeed, the complex transformation of agglomerated \ci{} DBs as suggested by results of the low C concentration simulations could involve an intermediate amorphous phase probably accompanied by the loss of alignment with respect to the Si host matrix.
202 % perfectly explainable by Cs obvious hkl match but not for DBs
203 In any case, the precipitation mechanism by accumulation of \cs{} obviously satisfies the experimental finding of identical \hkl(h k l) planes of substrate and precipitate.
205 % no contradictions, something in interstitial lattice, projected potential ...
206 Finally, it is worth to point out that the precipitation mechanism based on \cs{} does not necessarily contradict to results of the HREM studies~\cite{werner96,werner97,lindner99_2}, which propose precipitation by agglomeration of \ci.
207 In these studies, regions of dark contrasts are attributed to C atoms that reside in the interstitial lattice in an otherwise undisturbed Si lattice.
208 The \ci{} atoms lead to a local increase of the crystal potential, which is responsible for the dark contrast.
209 However, there is no particular reason for the C species to reside in the interstitial lattice.
210 Contrasts are also assumed for Si$_{\text{i}}$.
211 Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.
212 Until then, however, these may likewise be composed of stretched SiC structures coherently aligned to the Si host together with \si{} in the surrounding or of already contracted incoherent SiC surrounded by Si on regular lattice sites as well as in the interstitial lattice, where the latter is too small to be detected in HREM.
213 %In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
215 To conclude, results of the present study indicate a precipitation of SiC in Si by successive agglomeration of \cs.
216 Elevated temperatures result in increased entropic contributions to structural formation.
217 Moreover, conditions prevalent in IBS deviate the system from thermodynamic equilibrium.
218 Thereby, C$_{\text{i}}$ is enabled to turn into C$_{\text{s}}$ accompanied by the emission of Si$_{\text{i}}$.
219 \si{}, which is likewise existent, serves several needs: as a vehicle to rearrange the \cs{} atoms, as a building block for the surrounding Si host or further SiC and for strain compensation.
220 The \si{} vehicle turns \cs{} into highly mobile \ci.
221 This way, C can be easily rearranged in order to end up in a configuration of C atoms that occupy substitutionally the lattice sites of one of the fcc lattices of the diamond structure.
222 Stretched SiC structures arise, which are coherently aligned to the Si matrix.
223 \si{} is believed to likewise compensate the tensile strain within these structures.
224 This is followed by precipitation into incoherent 3C-SiC once the strain energy of the coherent structure surpasses the interfacial energy of the incoherent precipitate and the c-Si substrate.
225 The associated volume reduction is compensated by \si{} that may serve as a supply for further SiC or as a building block for the surrounding Si host and likewise reduce existing strain in the interface region.
227 Results of the atomistic simulation study that indicate the respective precipitation mechanism conform well with other experimental findings.
228 By verification, the derived conclusions with respect to the precipitation mechanism are reinforced.
229 Furthermore, experimental results that suggest a precipitation mechanism based on the agglomeration of \ci{} do not conflict with the proposed model of precipitation as concluded in the present study.
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