3 Combined ab initio and classical potential simulation study on the
4 silicon carbide precipitation in silicon
5 by F. Zirkelbach, B. Stritzker, K. Nordlund, et al.
10 First-principles study of defects in carbon-implanted silicon
11 by F. Zirkelbach, B. Stritzker, J. K. N. Lindner, et al.
17 thank you for the feedback to our submission.
19 > We look forward to receiving such a comprehensive manuscript. When you
20 > resubmit, please include a summary of the changes made, and a detailed
21 > response to all recommendations and criticisms.
23 We decided to follow yours and the referee's suggestion to merge the
24 two manuscripts in a single comprehensive manuscript.
26 Please find below the summary of changes and a detailed response to
27 the recommendations of the referee.
29 Most of the criticism is pasted from the previous review justified by
30 the statement that we did ignore or not adequatley respond to it.
31 However, we commented on every single issue and a more adequate
32 answer is hindered if the referee does not specify the respective
33 points of criticism. Thus, some part of the response might be
34 identical to our previous one.
41 --------------- Response to recommendations ----------------
43 > I am not happy with these two papers for a multitude of reasons,
44 > and I recommend that the authors rewrite them as a single longer
45 > paper, to eliminate the criticism of serial publication. I do not
46 > accept the authors argument that they should be two papers they
47 > address the same issues, using the same methods. If they were to
48 > be split into two papers, it would be one for the VASP
49 > calculations, and one for the MD this is not how I suggest you
52 We now combined the two manuscripts to a single comprehensive one.
54 > do it, though. First, though, the following issues should be
55 > addressed (some are simply pasted from my previous reviews, where
56 > I feel that the authors have ignored them, or not responded
59 > 1. I feel that the authors are a bit too convinced by their own
60 > calculations. They do not state the error bars that would be
61 > expected for calculations like this +/- 0.2 eV would be a very
62 > optimistic estimate, I suggest. That being so, many of their
63 > conclusions on which structure or migration routes are most
64 > likely start to look rather less certain.
66 In literature, very often, differences less than 0.2 eV are obtained
67 in DFT studies and respective conclusions are derived. For instance,
68 differences in the energy of formation ranging from 0.05 - 0.12 eV are
69 considered significant enough to conclude on the energetically most
70 favorable intrinsic defect configurations in Si (PRB 68, 235205
71 (2003); PRL 83, 2351 (1999)). This is due to the fact that existing
72 errors are most probably of the systematic rather than the random
73 type. The error in the estimate of the cohesive energy is canceled out
74 since it is likewise wrong in the defect as in the bulk configuration,
75 which are substracted in the expression of the defect formation
76 energy. Even if the defect formation energy is overestimated due to a
77 too small size of the supercell resulting in a non-zero interaction of
78 the defect with its images, this is likewise true for other defects.
79 Although the actual value might be wrong, observed differences in
80 energy, thus, allow to draw conlcusions on the stability of defect
81 configurations. This is also valid for diffusion barriers, which are
82 given by differences in energy of different structures.
83 In fact, differences of 0.2 eV in DFT calculations are considered
84 insignificant when being compared to experimental results or data of
85 other ab initio studies. However, the observed differences in energy
86 within our systematic DFT study are considered reliable.
88 > 2. Why is 216 atoms a large enough supercell - many defect
89 > properties are known to converge very slowly with supercell size.
91 Of course, choosing a supercell containing 216 atoms constitutes a
92 tradeoff. It is considered the optimal choice with respect to
93 computational efficiency and accuracy.
95 We would like to point out that, both, single defects as well as
96 combinations of two defects were investigated in such supercells in
97 successive calculations.
99 For single defects, the size of the supercell should be sufficient.
100 This is shown in PRB 58, 1318 (1998) predicting convergence of the
101 vacancy in silicon - the defect assumed to be most critical due to
102 the flatness of the total energy surface as a function of the ionic
103 coordinates - for supercells containing more than 128 atomic sites,
104 where the defect formation energy is already well estimated using
105 smaller supercells of 64 atomic sites. Thus, convergence of the
106 formation energies of single defects with respect to the size of the
107 supercell is assumed.
109 A repsective statement was added (Change 3).
111 > They appear to be separating defects by as large a distance as
112 > can be accommodated in the supercell to approximate the isolated
113 > defects, but then they are only separated by a few lattice
114 > spacings from a whole array of real and image defects how does
115 > that compare with taking the energies of each defect in a
118 Again, we would like to point out that it is not our purpose to
119 separate defects by a large distance in order to approximate the
120 situation of isolated defects. However, we find that for increasing
121 defect distances, configurations appear, which converge to the
122 energetics of two isolated defects. This is indicated by the (absolute
123 value of the) binding energy, which is approaching zero with
124 increasing distance. From this, we conclude a decrease in interaction,
125 which is already observable for defect separation distances accessible
128 Nevertheless, the focus is on closely neighbored, interacting defects
129 (for which an interaction with their own image is, therefore, supposed
130 to be negligible, too). In fact, combinations of defects exhibiting
131 equivalent distances were successfully modeled in a supercell
132 containing 216 atoms in PRB 66, 195214 (2002). At no time, our aim was
133 to investigate single isolated defect structures and their properties
134 by a structure with increased separation distance of the two defects.
136 An explanation of the binding energy and the relation to the
137 interaction of defects was added (Change 8).
139 > 3. Constant pressure solves some problems, but creates others
140 > is it really a sensible model of implantation? What differences
141 > are seen for constant volume calculations (on a few simple
144 Differences are supposed to be negligible small since only small
145 changes in volume are detected. However, in experiment, substrate
146 swelling is observed. Thus, to allow for full relaxation, simulations
147 were performed in the NpT ensemble. However, for the above-mentioned
148 reason, no fundamental differences are expected for single defect
149 configurations in the canonical and isothermal-isobaric ensemble with
152 A respective statement was added to the methodology section
155 > 4. What method do they use to determine migration paths? How can
156 > they convince us that the calculations cover all possible
157 > migrations paths that is, the paths they calculate are really
158 > the lowest energy ones? This is a major issue there are a
159 > number of methods used in the literature to address it are the
160 > authors aware of them? Have they used one of them?
162 The constrained relaxation technique is used to determine migration
163 pathways. The method is named and a reference is given in the
164 methodology section. The method not necessarily unveils the lowest
165 energy migration path. The supposed saddle point structure needs to be
166 attested by investigating the vibrational modes. However, reasonable
167 results are obtained for the specific system. In fact, so far, the
168 best quantitative agreement with experimental findings has been
169 achieved concerning the interstitial carbon mobility (PRB 82, 094110
170 (2010)) utilizing the constrained relaxation technique. Thus, obtained
171 migration paths are assumed to be valid without investigating the
172 vibrational modes of every single supposed saddle point configuration.
174 For clarity we added a statement that, of course, the true minimum
175 energy path may still be missed (Change 7).
177 > 5. I have some serious reservations about the methodology
178 > employed in the MD calculations. The values given for the basic
179 > stabilities and migration energies in some cases disagree
180 > radically with those calculated by VASP, which I would argue
181 > (despite 4 above) to be the more reliable values. The main
183 Indeed, discrepancies exist. However, both methods predict the C-Si
184 100 DB configuration to be the ground-state structure. The
185 underestimated energy of formation of substitutional C for the EA
186 potential does not pose a problem in the present context. Since we
187 deal with a perfect Si crystal and the number of particles is
188 conserved, the creation of substitutional C is accompanied by the
189 creation of a Si interstitial. The formation energies of the different
190 structures of an additional C atom incorporated into otherwise perfect
191 Si shows the same ground state, i.e. the C-Si 100 DB structure, for
192 classical potential as well as ab initio calculations.
194 This is discussed in full detail in section V in the combined
197 > problems is the huge over-estimate of the C interstitial
198 > migration energy (a process which is at the heart of the
199 > simulations) using the potential used in the paper. I am not
200 > convinced that the measures they take to circumvent the problems
201 > in the method do not introduce further uncertainties, and I would
202 > need a bit more convincing that the results are actually valid.
204 We hope to be able to convince by responding to the following
205 statement of the referee.
207 > The authors' circumvention of this is to do the simulations at
208 > much heightened temperatures. However, this only gives a good
209 > model of the system if all cohesive and migration energies are
210 > over-estimated by a similar factor, which is demonstratably
211 > untrue in this case. For this reason, despite the reputation and
212 > previous work with Tersoff (and similar) potentials, the results
213 > need a critical scrutiny, which I am not very convinced by in
216 There is not necessarily a correlation of the cohesive and migration
217 energies. You can always add a constant to the cohesive energies of
218 respective structures. It is the difference in the cohesive energies
219 of structures within the migration path, which determines the
222 In fact, cohesive energies are most often well described by the
223 classical potentials since these are most often used to fit the
224 potential parameters.
226 The overestimated migration barrier, however, is due to the short
227 range character of the potential, which drops the interaction to
228 zero within the first and next neighbor distance using a special
229 cut-off function as explained in PRB 76, 224103 (2007). The
230 overestimated barrier and slightly different pathway (however,
231 starting and final configuration/orientation agree) is indeed
232 demonstrated for the carbon interstitial within the present study.
233 Since the reason of overestimation is inherent to the short range
234 potential, migration pathways among other configurations are
235 likewise overestimated.
237 Since most of the defect structures show atomic distances below the
238 critical distance, for which the cut-off function is taking effect,
239 the respective formation energies are quite well described, too (at
240 least they are not necessarily overestimated in the same way).
242 Thus, increased temperatures result in an increased probability of
243 transition. Obviously, this enables the structural transformation
244 into energetically less stable structures of substitutional carbon and
245 interstitial silicon that are observed in the high temperature
246 simulations. Being in nice agreement with experimental findings, these
247 results suggest the usage of increased temperatures to constitute a
248 necessary condition to deviate the system out of the ground state as
249 it is the case in the ion beam synthesis process.
251 A respective statement and a more detailed comparison with experiment
252 was added to the combined version of the manuscript (Change 22).
254 Again, we would like to repeat the arguments that legitimate the usage
255 of increased temperatures although cohesive and formational energies
256 are not ovrestimated in the same way than the migration barriers.
257 While the properties of some structures near the equilibrium position
258 are well described, the above mentioned effects increase for
259 non-equilibrium structures and dynamics. Thus, for instance, it is not
260 surprising that short range potentials show overestimated melting
261 temperatures. This is not only true for the EA but also (to an even
262 larger extent) for Tersoff potentials, one of the most widely used
263 classical potentials for the Si/C system. The fact that the melting
264 temperature is drastically overestimated although the cohesive
265 energies are nicely reproduced indicates that there is no reason why
266 the cohesive and formational energies should be overestimated to the
267 same extent in order to legitimate the increase in temperature to
268 appropriately consider the overestimated barrier heights for
271 Indeed the cut-off effect increases if the system is deviated from
272 equilibrium. Thus, to mimic IBS, a process far from equilibrium,
273 increased temperatures are exceptionally necessary if short range
274 potentials are utilized.
277 --------------- Summary of changes ----------------
279 Since the new manuscript is a combination of manuscripts BC11912 and
280 BA11443, the following summary of changes mainly contains the
281 construction of the new manuscript by text blocks of previous
282 manuscripts. Please let me know if a more detailed summary of changes
285 The title of the new manuscript is that of BC11912. Thus, stated
286 changes apply to this manuscript.
292 Change 1: added/merged parts of the Abstract of BA11443
294 from: These aime to clarify ...
295 until: Finally, results of the ...
297 Change 2: added/merged parts of the Introduction of BA11443
299 from: A lot of theoretical work has been done ...
300 until: However, investigations are, first of all, ...
302 from: By first-principles atomistic simulations ...
303 until: Furthermore, highly accurate quantum-mechanical ...
305 Change 3: convergence of BZ sampling and size of the supercell
307 -Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
308 -The defect structures and the migration paths have been modeled in
309 cubic supercells containing 216 Si atoms.
310 +To reduce the computational effort sampling of the Brillouin zone was
311 restricted to the $\Gamma$-point, which has been shown to yield
312 reliable results\cite{dal_pino93}.
313 +The defect structures and the migration paths were modelled in cubic
314 supercells with a side length of \unit[1.6]{nm} containing $216$ Si
316 +Formation energies and structures are reasonably converged with
317 respect to the system size.
319 Change 4: only small changes in volume
321 +The observed changes in volume were less than \unit[0.2]{\%} of the
322 volume indicating a rather low dependence of the results on the
325 Change 5: name algorithm used for structural relaxation
328 +Ionic relaxation was realized by the conjugate gradient algorithm.
330 Change 6: name reason for reservoir choice
332 +This corresponds to the definition utilized in another study on C
333 defects in Si\cite{dal_pino93} that we compare our results to.
335 Change 7: CRT not necessarily predicts the minimum energy path
337 +While not guaranteed to find the true minimum energy path, the method
338 turns out to identify reasonable pathways for the investigated
341 Change 8: added definition and explanation of the binding energy to
342 the methodology section
344 from: The binding energy of a defect pair ...
345 until: The interaction strength, i.e. the ...
347 Change 9: removed Results section
349 Change 10: added 'Comparison of classical potential and
350 first-principles methods' section
352 +In a first step, quantum-mechanical calculations of defects in Si and
353 respective diffusion processes are compared to classical potential
354 simulations as well as to results from literature.
355 +Shortcomings of the analytical potential approach are revealed and
356 its applicability is discussed.
358 Change 11: comprehensive Table including all defects and methods
360 Change 12: added text on unstable hexagonal Si defect for classical
361 potentials - necessary due to combination of manuscripts!
363 from: The hexagonal configuration ...
364 until: While not completely rendering impossible ...
366 Change 13: added configurations that require spin polarized
369 from: Instead of giving an explicit value ...
370 until: No other configuration, within ...
372 Change 14: 'Carbon mobility' section of BC11912 mapped to 'Mobility of
373 carbon defects' section
375 Change 15: added 'Quantum-mechanical investigations of defect
376 combinations and related diffusion processes' section
377 corresponding to 'Results' section of BA11443
379 Change 16: added 'Mobility of silicon defects" section from III A of
382 Change 17: added 'Summary' section from 'Discussion' section of
385 Change 18: relocate 'Excursus: Competition of C_i and C_s-Si_i' section
388 Change 19: section 'Classical potential calculations on the SiC
389 precipitation in Si' and respective glue text added
391 from: The MD technique is used to gain ...
392 until: The approach is follwed and, ...
394 content corresponds to 'Results' section of BC11912
396 Change 20: 'Summary' section added containing parts of 'Discussion and
397 summary' section of BC11912
399 Change 21: 'Conclusions' section added containing parts of the
400 'Discussion' section of BA11443 and the 'Discussion and
401 summary' section of BC11912
403 Change 22: more detailed comparison to experiment added
405 starting from: Moreover, results of the MD simulations ...
407 Change 23: 'Summary' section added containing parts of the 'Summary'
408 section of BA11443 and the 'Discussion and summary' section