# The skyrmion-bubble transition in a ferromagnetic thin film

### Submission summary

 Authors (as Contributors): Anne Bernand-Mantel · Laurent Ranno · Nicolas Rougemaille
Submission information
Date accepted: 2018-04-23
Date submitted: 2018-04-09 02:00
Submitted by: Bernand-Mantel, Anne
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Experiment
• Condensed Matter Physics - Theory
Approaches: Theoretical, Computational

### Abstract

Magnetic skyrmions and bubbles, observed in ferromagnetic thin films with perpendicular magnetic anisotropy, are topological solitons which differ by their characteristic size and the balance in the energies at the origin of their stabilisation. However, these two spin textures have the same topology and a continuous transformation between them is allowed. In the present work, we derive an analytical model to explore the skyrmion-bubble transition. We evidence a region in the parameter space where both topological soliton solutions coexist and close to which transformations between skyrmion and bubbles are observed as a function of the magnetic field. Above a critical point, at which the energy barrier separating both solutions vanishes, only one topological soliton solution remains, which size can be continuously tuned from micrometer to nanometer with applied magnetic field.

Published as SciPost Phys. 4, 027 (2018)

Dear Editor,
To answer to the referee comments, we have modified the manuscript as described in the list of changes. In addition to minor corrections, we have carried out complementary micromagnetic simulations wich has been included in the manuscript in section 4.4.
Anne Bernand-Mantel

### List of changes

list of changes:
1. (requested change 1) We have replaced the sentence “This expression reproduced the exchange energy obtained in Section 2.3 with a 3% maximum error without adding any fitting parameters.” By “This expression reproduce the exchange energy obtained in Section 2.3 with a 3% maximum error in the full radius range.”
2. (requested change 2) We have replaced the sentence “ We use an approximate expression for the Zeeman energy of the topological soliton which represents the Zeeman energy difference between a film with a uniform +Ms containing a magnetic cylinder of radius rand opposite uniform -Ms magnetisation, and the Zeeman energy of the uniform +Ms state.
3. (requested change 5) The beginning of the caption of figure 3 has been replaced.
4. (requested change 6) We have replaced the expression “spin rotation energy” by “topological soliton wall energy” which is defined in section 2.3 as “the topological soliton wall energy density is defined as the sum of the exchange, anisotropy and DMI energy densities”
5. (requested change 6) In section 2.4.2 we have suppressed “and is decreasing the energy cost of the spin rotation”
6. (requested change 6) Description of the total energy in section 2.4.5.
7. (requested change 6) “spin rotation energy” has been replaced by “topological soliton wall energy” in several places.
8. (requested change 7) We have replaced “above which” by “at which”. We have suppressed “is starting at the critical point ($D_{\mathrm{cs}}$,$H_{\mathrm{cs}}$) and”.
9. (requested change 8) We have replaced “Zone 1 starts along the blue line” by “Zone 1 starts for $D$ values larger than D_{\mathrm{min.}}=2\sqrt{A_{\mathrm{ex.}}K_{\mathrm{eff.}}}/”
10. (requested change 9) We have suppressed the sentences in Sec 2.4.4: “However, is limited as in this radius range $E_{\mathrm{dem.}}^{\mathrm{long\ range}}$ is at least one order of magnitude smaller in amplitude than $E_{\mathrm{exch.}}$, $E_{\mathrm{DMI}}$ and $E_{\mathrm{anis.}}$ (see Fig.\ref{fig:1}e) as also shown by B\"uttner et al. \cite{Buttner2017}. “ and modified the sentence in Section 4.3.2 “the Zeeman energy and its variations are always one to several orders of magnitude smaller than the total energy value and total energy variations in the $r<\Delta$ range.”
11. We have added the section 4.4 with a new figure where the micromagnetic simulations are presented

### Submission & Refereeing History

Resubmission 1712.03154v3 on 9 April 2018
Submission 1712.03154v2 on 15 December 2017

## Reports on this Submission

### Anonymous Report 1 on 2018-4-13 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1712.03154v3, delivered 2018-04-13, doi: 10.21468/SciPost.Report.416

### Strengths

1. The paper would be useful to the experimental community in their search for skyrmions in DM materials, and in guiding the fabrication of new such materials.

2. The presentation is clear and detailed.

3. The analytical model studied in this paper while being particularly simple, it does capture the features of a complicated phenomenon.

### Report

The analytical model studied in this paper, while being particularly simple, it does capture the features of a complicated phenomenon. That is probably the main strength of the paper.

The resubmitted version of the paper removes the weak points. I list below some minor comments on the changes submitted with the new version.

Weakness point 2:
It is an improvement of the paper that the authors have added results of numerical simulations and discussed the comparison with their analytical model.
This has verified, for example, that the details about the point Dcs, Hcs are given correctly qualitatively and to a fine approximation quantitatively.
I would like to make clear here that I actually considered there was no doubt that the overall picture from the analytical model is correct, thus I did not think that a full set of simulations would be necessary.
In any case, the authors have clarified the details of the phase diagram.

Weakness point 3:
The explanations of the authors' reply regarding the relation of their work to previous works is sufficient. I would consider it though better if the authors had included some of these explanations in the paper.
In any case, I understand that the Reports and the Replies are accessible, just like the paper itself, so this information will be available.

Requested change 4:
This was referring to Fig. 3 (not to Fig. 1 as I erroneously wrote).
I still think some of the entries are redundant. However, it is up to the authors to decide whether these rather help make their results clearer.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -