The skyrmion-bubble transition in a ferromagnetic thin film

We thanks the referee for his valuable comments and his interest for our work. Here is our response to the weaknesses and request changes as well as a list of changes. The references mentioned with a number in brackets correspond to the version of the paper read by the referee (v2). A revised version has been submitted to arxiv (v3).

Referee : 1.”Expression (9) is usually a good approximation for the energy, but it is not easy to know when this could break down. Despite the detailed explanations in the paper on the limitations of the approximation, there may be further unexpected cases that this breaks down.”
Authors: As mentioned by the referee we have discussed in details the limitations of the model. First, it is restricted to films thinner than the exchange length and to DMI values sufficiently large to ensure a Néel nature of the DW. In addition, the approximations used will have an impact on the estimation of the skyrmion and bubble equilibrium radius mainly due to the approximations on the long range demagnetizing energy and on the Zeeman energy. This error, as described in the text, is small in the r<$\Delta$ region, as both of these energies and energies variations are small, compared to the other energies and energies variations, in this range. The error is also negligible in the r>>$\Delta$ range as the demagnetizing energy and the Zeeman energy are well-approximated in this range. In the intermediate radius range the error on the topological soliton radius may be significant in particular if $\Delta$ is large (weak anisotropy) and if we are close to the critical (Dcs,Hcs) point where the radius presents a strong susceptibility to energy variations. Despite this potential errors in the topological soliton radius estimation, the general features of the skyrmion-bubble diagram described un the paper will not be modified, as confirmed by our complementary micromagnetic simulations that we have carried out close to the critical (Dcs,Hcs) point (see description in pargraph 2. below).

Referee : 2. “Α weak point of the paper is that the results are not compared against simulations. It would be desirable that the subtle regions of the phase diagram (the region of skyrmion and bubble having similar radii or merging, and the regions of skyrmion bursting, or bubble collapsing) be confirmed by simulations.”
Authors: We have carried out complementary numerical simulations, which are presented in the new section 4.4. The micromagnetic phase diagram close to (Dcs,Hcs) showed in Figure 4a can be compared to the phase diagram obtained with the analytical model in Figure 4b. The results are very similar to what we predict with our analytical model using the same parameters, but we observe a shift towards 8% larger D values as expected and discussed in section 4.3.1. This confirms that the approximations in our model leads to an overestimation of Dcs, however all the predicted behaviors are qualitatively the same. We would like to point out that the diagram in Figure 4a realized using Mumax3 took days to be calculated while the diagram in Figure 4b is calculated in milliseconds. In conclusion, we think that micromagnetic and analytical models are complementary approaches.
Referee: “For example, it appears, in the phase diagram in Fig. 3f, that the bubble exist almost down to external field H=0. As far as I know, existing numerical simulations do not confirm this."
Authors: While our topological soliton model shows that there is a solution almost down to H=0, we introduce in section 1.1 a limit in our model visible as a dashed line in figures 3 a,b,d and f. As described in section 4.1.1 this line correspond to the limit where the observation of an isolated topological soliton is very unlikely as the uniform ferromagnetic state is not the ground state anymore and stripe phase will appear (see also Kiselev. ref. 35). This may explains why bubbles at low magnetic field are not observed in the numerical simulations mentioned by the referee.
Referee: "Another example is zone 0 in Fig. 3b. It should be easy to check numerically whether this approximate result carries over to the full model. “
Authors: The absence of skyrmion for too low values of D has been observed in previous numerical simulations (Sampaio Nat. Nanotech. 8, 839 (2013), Rohart PRB 88, 184422 (2013),…)

Referee: 3. “Apart of all the above comments, the relation of the present paper to previous works should be clarified. (Indeed, a very good presentation of the literature is provided). Specifically, Ref. [46] claims to provide a "Full phase diagram of isolated skyrmions". Apart from differences in the presentation of the phase diagram, I cannot see what is the additional information provided in the present paper compared to the results presented in Ref. [46]. I would ask the authors to clarify this. “
Authors: We agree that our work is closely related to Ref.[46] from Büttner et. al. (Scientific Reports 8, 4464 (2018)), this is why we mention their work several times in the paper. Both of our model and the model from Büttner et. al. are analytical models which aims at describing all energy terms for the stabilization of skyrmions and bubbles. However, there are strong difference between both models. The analytical model from Büttner et. al., which is avaliable in their supplementary materials, involves the use of a large number of fitting constants and complicated fitting functions which present artificial non monotonous variations as attested by the fluctuations of the errors presented in figures S1-S4 in the supplementary. Considering this, it is difficult to say if the multiple extrema observed by Bütter et. al. are not artifacts of the model. Another major difference between our model and the model from Büttner et al. is that their model present divergence or strong errors which makes it unusable in the r/$\Delta$ < 0.88 range while our model does not present any divergence and can be used down to r/$\Delta \rightarrow$ 0. For example, in the model from Büttner et. al. the exchange energy described in supplementary S2.2 possess a 1/$\rho$ term which diverges for $\rho \rightarrow 0$ and the anisotropy energy becomes negative for $\rho \rightarrow 0$. Despite these issues, some main features predicted by our model are also present in the phase diagram from Büttner et. al. (Figure 2.c): above a D value close to 2 mJ/m$^2$ continuous transitions between bubbles and skyrmions are observed and below this value, discontinuous transition and a region of bistability is present. These features are very similar to what we report.

Requested changes
Referee: 1. “Eq. (5) is claimed to give a good fit "without adding any fitting parameters". Actually all numbers appearing in Eq. (5) are fitting parameters, that have already been given appropriate values.”
Authors: We agree with the referee and we have suppressed this sentence. (see list of change 1)
Referee: 2. “Eq. (8) is explained to give "demagnetising energy difference between a magnetic cylinder with uniform magnetisation pointing in one direction and the uniform ferromagnetic state". This is not a precise expression. What is probably meant by "cylinder" is a uniformly magnetized film with a cylindrical domain with inverted magnetization?”
Authors: We thanks the referee for this remark we have corrected the sentence. (see list of change 2)
Referee: 3. “In the beginning of Sec. 3 it is stated that "changes in the fixed parameters do not modify qualitatively the results presented here but rather shift the main features to different D values." This result would probably follow trivially if one would work in a nondimensional form of the equation.”
Authors: We agree with the referee, we wanted to work with dimensional parameters for two reasons : 1) to make our results directly readable for experimentalist who are familiar with the parameters we use which are typical of thin films with perpendicular anisotropy. 2) as our programs are available to the reader it is easy to run the simulations with different parameters.
Referee: 4. “All the information in Figs. 1a,b,d,e seems to be contained in Fig. 1f. What is there in a,b,d,e that is not contained in Fig. 1f?
All the information in Figs. 1c seems to be contained in Fig. 1g.
If this is correct then Fig. 3 need to contain only entries f and g. »
Authors: We are not sure about which figures the referee refers to as there is no Figure 1f nor 1g. We agree that some informations are represented in two different ways in some figures for example in Fig. 1d and e. The aim of Fig. 1d is to emphasis the deviation of $\sigma_s$ compared to $\sigma_w$ due to the bending of the domain wall in the (x,y) plane.
Referee: 5. “In the figure captions (e.g., Fig. 3 and 4) it is, in same cases, not clear which entry the description is referred to. It would be clearer if each entry is explained separately.”
Authors: We have improved the figure caption (see list of change 3)
Referee: 6. “In many places the expression "spin rotation energy" is used. This expression is not a clear one.”
We agree this point was unclear. We clarified this. (see list of change 4-7)
Referee: 7. “In Sec. 4.1.1 (and in various other places) there is the expression "a blue dot in Fig. 3b above which...". Referring to a "region" above a "point" is not a clear specification.”
Authors: We clarified this. (see list of change 8)
Referee: 8. “In Sec. 4.1.2 we read "Zone 1 starts along the blue line". It is not clear what is meant.”
Authors: We clarified this. (see list of change 9)
Referee: 9.” In Sec 2.4.4 and 4.3.2 it is claimed that errors are negligible because some energy term is small compared to other terms. But, comparing energies is not safe. What matters is rather compare variations of energies.”
Authors: We agree, we have corrected this. (see list of change 10)