Typically, the domains are much larger than skyrmions. Furthermore, the skyrmions are more exposed to external electric fields

skyrmions are stablized because of the chiral DM interaction then in a sample with such interaction all of skyrmions have only one topological charge while bubbles are stablized usually because of non-chiral interactions such as dipolar interaction then in the sample bubbles with different chirality exist and they are degenerate.

A bubble is magnetized nearly uniformly while skyrmion's magnetization, namely the component perpendicular to the layer, varies considerably, with 'singular point' at its center.

Sic! That's right, Marek. But the question remains, how the inner structure of the magnetization affects the stray fields of the domain and skyrmion outside the plate (above its surface), so that it can bedetected and distinguished with the help of the MSM. Perhaps, the distinction can be perceived (sensed) due to the tangential component or by stray fields of the domain wall of magnetic bubble.

Waiting for a response from Kai Fauth; he knows the answers to all the questions.

The line between a skyrmion and a bubble is a very debatable topic nowadays.

Perhaps, one could use a large number of defects (bloch lines) as an argument for calling them as bubbles, since when such structures are large, they indeed tend to easily change their topological number due to such defects. It might be difficult to detect these defects with MFM, which is only mz-sensitive, but might be doable with in-plane sensitive Lorentz microscopy.

However, a recent trend is to call everything from this family as just a "skyrmion". There are valid arguments to do so.

First, skyrmions and magnetic bubbles are topologically equivalent (assuming they hold the same topological charge), so it would be hard to distinguish between these two, unless one uses (very arguable) distinctions that exist in literature since the 90s, such as their size, field-stabilization, the relation between the domain wall width and skyrmion radius, or how many spins are exactly out-of-plane inside of a skyrmion.

Additionally, it has been recently confirmed that small and stable skyrmions can also exist in the classical "bubble" domain: not only in the absence of external magnetic fields, but also in the absence of DMI (purely due to the stray field interactions, seeArticle Full phase diagram of isolated skyrmions in a ferromagnet

See the new paper:

What makes magnetic skyrmions different from magnetic bubbles ?

by Andrei B. Bogatyr\"ev and Konstantin L. Metlov, 5 pages 4 figures

available at http://arxiv.org/abs/1711.07317 , 3490kb

Ivan, you did refer to the manuscript by Buettner et al. ¨Full phase diagram of isolated skyrmions...¨. However, the authors of the manuscript even did not write the energy functional they used for calculations/simulations. Probably, it has lower generality than the authors claimed. From the other side, I am sure that the authors used some particular ansatz for the skyrmion magnetization distribution (it also not shown in the manuscript). Therefore, the phase diagrams by Buettner et al. (Fig. 4) are plotted within some approximations of unclear accuracy.

Konstantin, yes, Buettner et al. was indeed one of the papers that I referred to. In this paper, the circular symmetric 360-wall profile has been used for the energy functional. This theoretical profile has been found to fit well not only the simulations data (there, the equilibrium profile is achieved by the relaxation procedure), but also the experimental data for small skyrmions from various papers.

Ivan, what experimental data do you mean? What kind of the skyrmion profile? Is it the same as Rohart and Thiaville [PRB 2013] did use?

Konstantin, the proposed in Buettner et al. theory was based on the 360 wall skyrmionic profile (eq. 3.15, 3.17 in Braun [PRB 1994]), and it was tested on the experimental profile by Romming et al. [PRL 2015]. As for the Rohart and Thiaville [PRB 2013] that you mentioned, they only used a 180 degree wall profile (Eq. 9 in their paper with r instead of x). The 360 wall profile that was used in Buettner et al. is a more general case, which of course, reduces to this 180 degree profile for large skyrmions.

By the way, there is a fresh article about skyrmion-bubble transition:

https://www.researchgate.net/publication/321670473_The_skyrmion-bubble_transition_in_a_ferromagnetic_thin_film

Ivan, any Bloch or Neel skyrmion in 2D ferromagnets (thin magnetic films or dots) is 180-degree domain wall if considered as a function of the radial coordinate r. The polar magnetization angle \Theta(r) changes from 0 to \pi (or from \pi to zero). Namely such magnetization profile corresponds to the skyrmion topological charge Q=1. Braun [PRB 1994] considered 360-domain walls or domain wall pairs in 1D ferromagnets (long wires). The type of domain wall (180 degree) does not depend on the skyrmion size. Romming et al. [PRL 2015] also observed 180-degree radial domain walls of nanosize skyrmions. Although, any 1D cross section of the skyrmion profile along in-plane direction formaly looks as 1D 360-domain wall, we should not forget that indeed we deal with 180-degree radial domain wall in 2D ferromagnet.

Alexander, thanks for the reference to the manuscript by Bernard-Mantel et al. ¨The skyrmion-bubble transition in a ferromagnetic thin film¨.

Konstantin, 

Actually both Romming et al.  [PRL, 2015, Eq. 1] and Boulle et al. [Nature Nano 2016, the discussion below Eq. 1] used the 360 wall solution as a fit for their experimental data.

Formally, 180 degree-wall is the solution only for the single straight domain wall. For small skyrmions, there is no analytical 180 degree-wall-like minimizer of the energy functional. However, in order to find an appropriate approximation for small skyrmions, one has to use something like a 360 wall profile, because just like in the case of the 1D 360 walls, there exist overlapping tails from the domain walls located in the opposite sides of the skyrmion, which are nicely taken care by this 360-wall treatment. In contrast, the 180 degree wall profile fails to take this overlap into account, and for that reason, it is only valid for large skyrmions (when the overlap of the tails is negligible). 

Ivan,

When you talk about overlapping tails from the domain walls located in the opposite sides of the skyrmion, do you refer to long range demagnetising (=stray field =SF) effects? I am interested to know the impact of this SF effect on very small skyrmions for very thin films has it has been argued that they are negligible [ex. Leonov NJP 2016].

Concerning Butter paper "Full phase diagram of isolated skyrmions in a ferromagnet" I am really looking forward to read the full version as there is no much details on the exact model in the preprint, as Konstantin said.

In the mean time, I am wondering if, the fit of a skyrmion radius vs field (as in Romming et al. [PRL 2015] and Buttner et al.) is really a good argument? Isn't it that:

1) there is too many parameters in these models (Ku, t, Ms, D). Using classical model in Romming's paper, I can obtain very close "size versus magnetic field" dependences and skyrmion profile with different sets of parameters. Isn't it a typical issue often mention as "fiting a elephant" in reference to John von Neumann?

2) As Romming fits his skyrmion size vs field dependence with a model neglecting the SF effect and Buttner fits it including the SF : either, it means that the stray field effect is negligible for this type of skyrmion, or, this is only due to the reason in 1), no?

Anne,

I started to read your manuscript on the magnetic skyrmions in thin films [arXiv_condmat_Dec 08, 2017]. Is my understanding correct that what you did is just fitting of the different simulated contributions to the skyrmion energy by simple analytical functions of the skyrmion radius (R_s)? Can we say that the radius R_s is sufficient to describe the skyrmion energetics and stability?

Yes, this is what we do at the beginning of the paper, we show the agreement between the energy contributions from the fonctionnal and our analytical expressions. However there is no fitting parameter in the model, it derivre from simple considerations implying approximations , mainly on the long range demag and zeeman for wich we considered a cylinder of unifrom magnetization.

The radius may give an idea of the skyrmion energetics and stability for low radius skyrmions (typically

Anne,

thanks for your explanations.

I have several questions/comments to your manuscript (ArXiv:1712.03154v1):

1) there is a minimal Dzyaloshinskii-Moriya interaction strength D_min in your phase diagram, Fig. 3b. D_min is approximately equal to (2/pi)*sqrt(A*K). Can we say that existence of the finite D_min contradicts to the phase diagram by Kiselev et al. [J. Phys. D 44, 392001 (2011, Fig. 3)], where an isolated skyrmion in infinite films is metastable down to D=0?

2) the skyrmion/bubble bi-stability (or simultaneous existence of two energy minima corresponding to small and large radius solitons) was already calculated for ultrathin dots Ir/Co/Pt by M. Zelent et al. Phys. Stat. Sol. - RRL 11, 1700259 (2017). There is no principal difference between the dot with in-plane size 500 nm and infinite film.

Konstantin,

I just noticed you asked further questions. Thank you for the interesting questions. I will answer them now but maybe it will be interesting for people if you post them as well as a comment to our paper here: https://scipost.org/submissions/1712.03154v2/

1) When the skyrmion energy functionnal (Eq. (3) in Kiselev work, Eq. (1) in our work) is minimized, a skyrmion profile minimizing the energy is found down to vanishing values of D. However this does not mean that the energy minimum is a metastable state. Indeed such a minimum is found even for D=0, a case where it is well known that there is no stable solution (Derrick J.of Math. Phys. 5 1252 (1964)). This simple exemple illustrate that the obtention of a solution with this method does not garanty that it is a stable solution. Recent works have tryed to estimate the energy barrier protecting the skyrmion using atomic spin models (ex. Sci. Rep. 7(1), 4060 (2017)).

In our paper we use a simple mathematical considerations to deduce that there is a minimum D to stabilize skyrmions in the framework of our model. In a real sample however there is some other potential energy terms that could stabilize skyrmions for low D or in absence of D such as higher order exchange terms for example.

2) The interesting result form M. Zelent et al. also evidence a bistability and we will cite this paper in our corrected version

Anne,

I just found your manuscript published in SciPost Phys. 4, 027 (2018). I did not see there essential changes in comparison with the version dated Dec. 8, 2017, we discussed.

I did not understand you answer to my question concerning contradiction between your skyrmion stability phase diagram and the diagram by Kiselev et al. [J. Phys. D 44, 392001 (2011, Fig. 3)] in the limit D(DMI parameter) goes to zero.

You cannot refer to the Derrick theorem because Derrick et al. considered soliton stability in 3D infinite sample. You deal with infinite film, i.e., you have two surfaces restricting your sample and should account for some surface contributions to the magnetic energy.

The best,

Konstantin.

Anne,

some other comments to you paper A. Bernand-Mantel et al., SciPost Phys. 4, 027 (2018).

Sec. 2.4.4. Demagnetising energy.

You wrote that ¨demagnetising energy cannot be expressed analytically and some approximations have to be used¨ referring to the paper by Y. Tu, J.Appl.Phys. 42, 5704 (1971). This statement is incorrect in application to the radially symmetric skyrmions (vortices, bubble domains). Y. Tu supplied exact expression for the soliton magnetostatic energy for the case of infinite film of finite thickness. Me with co-authors wrote exact expressions for the magnetostatic energy of the Bloch skyrmions [K.Y. Guslienko, IEEE Magn. Lett. 6, 4000104 (2015), K.Y. Guslienko and Z.V. Gareeva, J. Magn. Magn. Mat. 442, 176 (2017) ] and the Neel skyrmions [K.Y. Guslienko and Z.V. Gareeva, J. Magn. Magn. Mat. 442, 176 (2017)] in cylindrical dots. These expressions account for all magnetic charges: face surface, side surface and volume charges. See Eqs. (7), (7´) in the latter paper.

Therefore, I do not understand why in Eq. (8) you did use the magnetostatic energy of the uniformly magnetized cylinder taken from the textbook by Hubert et al.? The expression (8) neglects the volume magnetic charges, which are essential for the Neel skyrmions, and assumes infinitely thin skyrmion width. Both approximations are not realistic.

I also attract your attention to my recently published paper [K.Y. Guslienko, Appl. Phys. Express 11, 063007 (2018)], where I calculated stability diagram of the Neel skyrmions in ultrathin cylindrical dots.

The best,

Konstantin.

Konstantin,

Thank you for your comments and references. The main difference in the SciPost version is the comparison with Mumax Micromagnetic calculations we added in section 4.4.

Concerning your previous question about the contradiction between our skyrmion stability phase diagram and the diagram by Kiselev et al. [J. Phys. D 44, 392001 (2011, Fig. 3)] in the limit where D goes to zero. I agree there is a difference as we predict no stable skyrmion solution for D0) where only exchange and anisotropy are taken into account. Our predition of a minimum D value is interesting and in agreement with other calculation (see Rohart PRB 88, 184422 (2013) Fig. 4). It also seems to be in agreement with your recent calculations (Fig2 Guslienko APEX11 063007 2018 "The Néel skyrmion is unstable for small D according to Fig. 2").

Thanks for your remark about the demagnetising energy, I agree my sentence is rather unclear. Of course one can write the full exact energy functional as you mention. But while the exact expression of the soliton energy functional is know, approximations has to be used to obtain the solution for the infinite thin film case as in Tu's paper. The case of dots as described in your work is more favorable as long range effects are partialy suppressed and one does not have to deal with numerical intergration up to large radial distances as for the case of infinite films.

Concerning your remark on equation (8). The local approximation for the magnetostatic energy that we describe in section 2.2 (also used in Guslienko APEX11 063007 (2018) is based on the fact that volume charges are negligible compared to surface charges as far as I understand. Of course for a Néel skyrmion the presence of volume charges represent an energy cost but this contribution vanishes for thickness much smaller than the Bloch width (pi.sqrt(A/Keff)). Concerning the use of the textbook thin wall formula to describe the long range demagnetising effects we agree it is not realistic for skyrmions smaller than the Bloch width as discussed in section 4.3.1. Consequently our predictions about the skyrmion-bubble transition have to be confirmed by more accurate calculations.

Best

Anne

Hi Ankit,

In order to define a magnetic structure, we have done some experiments and confirmed the experiments by the simulations. I hope this helps : https://www.researchgate.net/publication/319642731_Observation_of_Magnetic_Radial_Vortex_Nucleation_in_a_Multilayer_Stack_with_Tunable_Anisotropy

Ali Taha Habiboglu

Ali,

it is difficult to extract information about internal magnetization structure of Neel skyrmions (radial vortices) from your MFM images shown in Figs. 4, 5 of your paper.

Anyway, I suggest do not use the termin ¨radial vortex¨ because ¨vortex¨ in its initial sense means a magnetization curling (flux closure, swirl etc.) magnetization configuration with mainly azimuthal component of in-plane magnetization. See, for instance, my review paper [K.Y. Guslienko, J. Nanosci. Nanotechn. 8, 2745-2760 (2008)] and references therein.

Konstantin.

Dear Ankit, dear all,

nice discussion. After seeing some works by Prof Hans J. Hug, I would say that the answer is MFM is able to do that, ...if you are skillful enough.

https://www.empa.ch/web/s203/quantitative-mfm (there might be better examples, I was looking for some vector stray field mapping publication but could not find it...but he has done)

It will be very nice to see if we can have an answer from the very expert...

Manuel