Peer review of: Spontaneous creation and annihilation dynamics and strain-limited stability of magnetic skyrmion  (2020)

Reviewers' Comments: 
Reviewer #1: 
Remarks to the Author: 

In their article „Spontaneous creation and annihilation dynamics and strain-limited of magnetic 
skyrmions” by F. Rendell-Bhatti and co-workers stability and transformations of skyrmions in the 
domain boundary region between two skyrmion grains in the prototypical skyrmion hosting B20 
material FeGe is investigated by Lorentz transmission electron microscopy (LTEM). 
 
The author focus on the continuous transformation of the position of individual skyrmion tubes in a 
particular 5-7 defect in the grain boundary. Here, the skyrmion arrangement is obviously not in its 
ground state and rearrangement of the skyrmion lattice is observed. Using a high speed detection 
system, the authors investigate the occurring configurational states, their spatial arrangement 
(also in comparison with a stable 5-7 defect), their statistics and their lifetime distribution. Using 
the simple Arrhenius law (and assuming that the attempt frequency is 10^9 Hz) the extract 
energy barriers for the spatial rearrangements. From their measurements and comparison to 
micromagnetic simulations they also infer a creation and annihilation mechanism which is 
reminiscent to the vortex core mechanism of magnetic vortices (not cited). 
 
The paper is well written, and easy to follow. It relies on some assumptions (e.g. the attempt 
frequency mentioned above or the assumption that the short-lived Tx states can be somewhat 
captured with the temporal resolution available in the experiments). 
However, I am very doubtful if an essentially 2D simulation of the skyrmion lattice is sufficient to 
capture the underlying physics here. Important, I guess, is the transition of the skyrmion tubes to 
new positions and this can initialize at various positions within the tube. What is more, the 
simulations are performed at zero temperature where the skyrmion tubes are static. In reality the 
wiggle around which can be seen in the experiments and motivates using the Arrhenius law. 
Without the comparison to micromagnetics an interpretation soley based on the LTEM images is 
hardly possible. 
It is for this reason that I would like to ask the authors to consider more realistic simulations 
before resubmission to Nature Communications. 
 
 
 
Reviewer #2: 
Remarks to the Author: 
This work is tackling the problem of topology concerning generation and annihilation of magnetic 
skyrmions which is of great interest in the field of magnetics. The authors investigated the 
dynamics of skyrmions at grain boundary of skyrmion crystals (SkX). They carried out the direct 
observation by taking Lorentz TEM movies of skyrmions in a thin plate of B20-type FeGe and 
extracted statistical information about dynamics of skyrmions through a detailed analysis on the 
Lorentz TEM movies. They also carried out micromagnetic simulations to support the observed 
dynamics of skyrmions. They demonstrated that the experimentally observed dynamics is 
governed by thermal activation in a system with energy landscape related to strain of magnetic 
structure. This finding provides an important clue to understand the structural stability and the 
topological aspect of skyrmions, and can be utilized for controlling skyrmions through strain of 
magnetic structure. 
 
Overall, the experiments, analysis, and simulation are clear and of high quality, and the discussion 
provides detailed explanation and interesting insight. From physical point of view, however, there 
are some unclear points. Before a decision can be taken, I would like to give some comments to 
check scientific accuracy and (hopefully) improve the manuscript: 
 
1) I feel Fig. 3b misleading because the colored broken lines are shown at the tops of the potential 
and not at the bottoms of the potential which is corresponding to the states. 
 
2) The authors show asymmetric potential barriers in Fig. 3b. According to it, for example, the 
energy barrier from P1 to T1 is lower than that from T1 to P1. This should result in lower transition 
probability for transition from P1 to T1 than that for transition from T1 to P2. However, the authors 
states at p.4, l.32 “Figure 3c also demonstrates the symmetry of the transitions, where transitions 
to or from a particular state had equal probabilities”, which seems to contradict with the transition 
probabilities expected from the potential barriers. In my opinion, the symmetry of Fig. 3c is just 
due to appearance and due to the normalization. The authors extracted transition events so that 
total probability becomes 1. In the present figure, each row is multiplied by the frequency of each 
state. I suggest normalizing the probability for each row. In addition to it, I think the transition 
events can be modeled using Markov chain, and I also suggest providing the probability of self-
transition, i.e. transition from a state to the same sate, as well. 
 
3) In Fig. 4a and 4b, the authors show images of a stable 5-7 defect with explanation of “a static 
5-7 defect from a different SkX”. In Fig. 4c shows “A Fresnel image of a stable5-7 defect (from the 
same SkX domain boundary as the dynamic 5-7 defects)”. However, the temperature and 
magnetic field is not shown explicitly for these images. Is the temperature and the magnetic field 
is the same as the other figures (253 K, 510 Oe)? If so, what is the difference of stable 5-7 defect 
and dynamic one? 
 
4) In Fig. 4d and 4e, the authors show that the split and merge of skyrmions can be explained in 
terms of creation and annihilation of antiskyrmion localized at the middle of cores of the two 
skyrmions. This implies the energy barrier required for the transition takes place by two steps: 
first step is the deformation of skyrmions (the first red arrow) and the second step is the 
annihilation of antiskyrmion structure (the second red arrow). The first step seems to be clearly 
reasonable and to be achieved only by continuous deformation. The second step, however, the 
transition from the middle state of Fig. 4d and 4e to the last state is not discussed in the 
manuscript although it is not clear. Concerning this, I found that the only the difference between 
the middle state of Fig. 4d and 4e is direction of the magnetic moment of the antiskyrmion core. 
After that, I came to think the transition path from Px to Tx might be like this: Fig.4d initial state -
(continuous deformation)-> Fig.4d middle state - (antiskyrmion core flip)-> Fig.4e middle state -
(continuous deformation)-> Fig. 4e initial state. And, the transition path from Tx to Px is the 
opposite. However, I am confused because the flip of the magnetic moment at aitskyrmion core 
changes the topological chrage by 2, which cannot meet the consistency of the topological charge. 
Do the authors have any comment of my consideration or any clear answer about the second 
step? 
 
5) I understand that even single temperature is enough for just comparing relative energy barrier 
scales if the observed events obeys the Neel-Arrhenius equation. However, this paper does not 
provide clear information that the observed events really obeys the Neel-Arrhenius equation. I 
think it would be better to include data at other several temperatures to support this if possible. 
Do the authors have any data to support or any comments about this suggestion? 
 
6) The present analysis was done on the specific SkX grain boundary. In general, not limited to 
magnetic skyrmions, there are many types of grain boundaries of crystals due to many degrees of 
freedom, such as relative orientation of crystals and rigid body translation, and the stability and 
the properties differ depending on the grain boundaries. I think this also applies to two-
dimensional SkXs. How can the present finding be generally applicable for the general grain 
boundaries of SkXs, for example, ones with a different misorientation angle? 
 
7) Typo: an extra “regular” in caption of Fig. 2, l.6. “skyrmions of regular regular six-fold 
coordination”. 
 
Overall, the subject and experimental results of this work is exciting and has enough potential for 
publication in Nature Communications if the points discussed above have been dealt with correctly. 
 
 
 
Reviewer #3: 
Remarks to the Author: 
In this study, the authors report the observation of spontaneous creation and annihilation of 
skyrmions in regions of high exchange energy density as the domain boundary of the skyrmion 
lattice in the FeGe thin film, using Lorentz TEM with a high-speed imaging detector, and Lorentz 
DPC STEM. The distinct states are observed in dynamic 5–7 defect. The authors discuss that the 
spontaneous transition occurs via unstable antiskyrmions in the light of skyrmion stability and 
topological charge conservation. Although the transition is quite interesting and had not been 
performed experimentally in the thin films, I am hesitating to recommend publication of this 
manuscript in Nature Communications for the following reasons. 
 
First, it is difficult to understand for me the intention of Figs. 4a-c (the DPC and Lorentz TEM 
images) to support the author’s case as the main result where the creation and annihilation of 
skyrmions are mediated by antiskyrmions. The reason is that there is no clear description of the 
observed stationary states such as Px, Tx, intermediate state between Px and Tx. In addition, 
there is no description how the stationary images were experimentally obtained. Lower 
temperature than 253 K in Figs. 1, 2? Otherwise, if the stationary images were obtained at 253 K, 
the high-resolution DPC image makes no sense because of magnetically varying in time during 
scanning of electron beam. 
 
Second, this study lacks the confirmed experimental evidence of the existence of both the 
elongated skyrmions (left in Fig. 4d and right in Fig. 4e) and the adjacent skyrmions completely 
divided into two (right in Fig. 4d and left in Fig. 4e) as the energy minimum states under thermal 
fluctuation. The reasons are as follows. As shown in attached Figs. 4a-c that add my comments, in 
the DPC images (Figs. 4a, b) the SkCoH is no longer single elongated skyrmion because of 
merging with the adjacent SkCoP and skyrmion. (As mentioned above, it is difficult to understand 
for me the intention of Figs. 4a-c.) On the other hand, all Lorentz TEM images cannot resolve the 
above merged state, as exemplified in Fig. 4c. Despite this Lorentz TEM resolution limit, the 
authors presuppose the existence of both the elongated skyrmions and the adjacent skyrmions, on 
the basis of the Lorentz TEM images obtained under thermal fluctuation. I think that it is necessary 
to clearly show the DPC image of the single elongated skyrmions without merging and the 
adjacent skyrmions completely divided into two. 
 
Third, as related to the above mentions, the antiskyrmion-mediated creation and annihilation of 
skyrmions as illustrated in Figs. 4d, 4e cannot be interpreted by the experimental figures only. 
Basically, these behaviours could not be directly observed by the TEM experiment and are 
supported by the micromagnetic simulation described in Supplementary Materials, as with Ref. 18. 
At least, I consider that it is better to put the micromagnetic simulation results in the manuscript. 
 
The other comments: 
 
In Figs. 4d, 4e, the spin textures surrounded by the red square are like antimerons rather than 
antiskyrmions. 
 
The discussion of 5–7 defects on page 5, lines 12-18 in the manuscript and on page 1, lines 21-28 
in Supplementary Materials seem to be incorrect. The authors describe that the total energy is 6% 
higher and unstable for a 5–7 defect because the energy of a five-fold coordinated skyrmion (5-
Sk) is 24% higher than that of a six-fold coordinated skyrmion (6-Sk) and the energy of a seven-
fold coordinated skyrmion (7-Sk) is 18% lower than that of the 6-Sk. However, in real FeGe thin 
films, the triangular skyrmion lattice is the most stable, i.e., the energy of the 6-Sk is the most 
minimum in the system. Therefore, when the skyrmion arrangement is deviated from the 
triangular lattice, the energy of such individual skyrmions should be always higher. Basically, the 
energy of the 5- and 7-Sks is always higher than that of the 6-Sk. Of course, the simulation should 
output the same results. The reason why the author’s micromagnetic simulation output the 
incorrect results is that the appropriate lattice constant is not set for the triangular skyrmion 
lattice. It has been reported that, in the micromagnetic simulation, the lattice constant of the 
triangular skyrmion lattice becomes larger than that of the triple-Q model: (4*pi*A/D)*2/sqrt(3) 
due to the influence of the magnetic and demagnetizing fields [K. Shibata et al., Phys. Rev. Lett. 
118, 087202 (2017)]. In the author's simulation, since the lattice constant of the triangular 
skyrmion lattice in the domain is smaller than the appropriate lattice constant, the integral energy 
value of the 7-Sk, of which the size is larger than that of the 6-Sk, is incorrectly lower than that of 
the 6-Sk. 
In addition, when comparing the energies of individual skyrmions, it should be compared not with 
the integral energy value per the skyrmion, but with integral energy value divided by the volume 
of the skyrmion (energy density). For example, if an elongated skyrmion has the same integral 
energy value with an unelongated (cylindrical or circular) skyrmion, the area of integration in the 
former is wider than that in the latter. Therefore, the absolute energy density value of the former 
is smaller than that of the latter. As the result, the former has higher energy than the latter 
because the calculated energy is negative. 
For the above reasons, the authors need to calculate again, and revise the description about 
comparing the energies. 
 
Part of the caption of Fig. S2 in Supplementary Materials is displayed by white characters.