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