Vortex states in a PbTiO$_3$ ferroelectric cylinder

Dear Editor,

please find the updated version of our manuscript, with comments from Reviewer 2 taken into account.

We thank the Reviewer for careful reading of the manuscript and helpful suggestions. We have implemented the requested changes and corrected the typos. Below, we provide a detailed point-by-point reply to all the comments made.

Answers to the Report 2 on 2022-9-19

1. Reviewer: In the case of a-vortices, how is the orientation of the in-plane vortex-axis determined? Is it artificially aligned with the [100] or [010] axis or can it take any other in-plane direction?

Answer: in our simulations, the system relaxes from the paraelectric state to $a$- (or $b$-) oriented vortex without any predefined vortex axis direction. If we detect several metastable states we select the one with the lowest energy. Therefore we are confident that we always obtain the stable state. In fact, the vortex core is aligned with either [100] or [010] axis due to the crystal symmetry, both $a$- and $b$-state being energetically equivalent. We added a note to Section 4.2.

1. Reviewer: p.4 When stating that the c-state occurs only in very long cylinders with h>500nm: shouldn't it also vary as a function of R?

Answer: indeed, the resulting state depends on the aspect ratio and thus varies as a function of $R$; namely, the $c$-phase region becomes shorter as the radius increases. However the further investigation of bigger cylinders is challenging in terms of ‘valuable result’ to ‘power consumption spent on calculation’ relationship. We have commented on this aspect in Section 3.

1. Reviewer: p.5 Figure 2 The solid blue curve visible on the phase diagram is explained only later (p.9, Fig.4 + text) - this information might be added already in the caption of Figure 2.

Answer: we thank the Reviewer for pointing out this issue. We have added the explanation in Section 3 and the caption of Fig.2 as suggested.

1. Reviewer: p.6 The authors mention that "the observed perturbed polarization texture at cylinder edges possesses the spontaneously broken chirality, the effect that was thought to occur due to the flexoelectric contribution" - it would have been nice if the authors extended a bit this very interesting discussion. Indeed from Fig. 7, one sees that the strain gradients are very large, therefore one expects an important role played by flexoelectricity.

Answer: we have run a series of additional calculations with account of flexoelectric coupling term, $-\frac{1}{2}f_{ijkl}\left( P_{k}\partial_{l}u_{ij}-u_{ij}\partial {l}P\right) $, using the typical values of flexoelectric tensor components found in literature (Ponomareva et al, Phys. Rev. B 85, 104101(2012); Yudin and Tagantsev, Nanotechnology 24, 432001 (2013)). We don’t observe any significant impact of flexoelectricity on the resulting state, i.e. no new states in addition to those shown in the phase diagram (Fig.2). For instance, Fig.A1 (see the link attached to the List of changes) demonstrates the comparison between the polarization and strain distribution in Fig.7 of the manuscript and the distributions obtained with account of the flexoelectric term. Although the strain gradients are indeed considerable, the typical coupling coefficients are small, which limits the role of flexoelectricity.

1. Reviewer: Since the authors are also experts in Hopfions, I would have liked a longer discussion on the differences and similarities between vortices and Hopfions and why the latter are not observed here. The authors mention the Arnold theorem, but this is not clear to me. Why is the anisotropy the key parameter? Shouldn't it be the geometry?

Answer: we are pleased to discuss the possibility of Hopfions emergence in ferroelectrics since, together with vortices, they form the building blocks of the topological ferroelectricity. In fact, the geometry, i.e. the geometrical confinement, enables the application of Arnold’s theorem to our system. According to it, the divergenceless vector field inside the restricted volume may swirl either in vortex or in Hopfion. Among these two, the more energetically favourable configuration is realised. In turn, the system energy is driven by the crystal anisotropy. As we already stressed in our previous publication, Ref.[30], the pronounced anisotropy of PTO favours the formation of vortices; however, in less anisotropic PZT the appearance of Hopfions is expected. Our current study on PTO cylinders confirms this statement: only more energetically favourable vortices are observed due to the relatively strong anisotropy. We expect, however, and this is confirmed by preliminary simulations, that Hopfions emerge in the similar geometry for PZT ferroelectrics. We have extended the discussion on geometry and anisotropy in Section 5.

1. Reviewer: p.8 "At larger radii, R>14nm..." this statement is valid for h = 6nm.

Answer: we observe that the fragmentation of the system on multiple vortices occurs at radii R>14nm, at h=6-12 nm, this is region VII in diagram in Fig.2. We have corrected the statement correspondingly.

1. Reviewer: Looking at Figure 7, as mentioned before, one sees a drastic spatial change of the strain components. This means that the PbTiO3 unit cells are drastically deformed, especially near the vortex core. The calculations are based on the bulk parameters. How valid is this for unit cells that are so much deformed? It would be nice if the authors could comment on that.

Answer: Indeed, within our approach the strain distribution close to the vortex core is challenging to calculate analytically, so we can reconstruct only the long-range asymptotic behaviour (15) as shown in Fig. 7b by dotted lines. This is a common methodology of the Ginzburg-Landau approach for calculations of vortices, for example in superconducting materials and superfluids, when the calculation of the vortex core energy is beyond the lowest-gradient-expansion approach and is treated as phenomenological parameter, using the extrapolation of the data from the low-gradient region. In our case the value of the unit cell deformation can be indeed underestimated, however, this does not affect the overall structure of the observed vortices at the large scale.

1. Reviewer: Figure 9b, one observes a clear deviation from the linear scaling for large R - any comment/explanation?

Answer: the scattered points in Fig.9 correspond to the a-vortex states close to region II (twisted DW state) in diagram in Fig 2, where the vortex core may be already distorted and thus the assumption given by (4) can deviate. We have added the explanation in Section A.3.

1. Reviewer: Typo p.3: reach -> rich

Answer: many thanks for careful reading. We have fixed the typo.

1. A note considering the a-vortex axis orientation added to Section 4.2.
2. Section 3 is expanded with the discussion on the size of $c$-phase region in longer cylinders (nanowires) and the explanation of the separation line in Fig.2 (also in the caption to Fig.2).
3. A comment on Hopfions and the role of anisotropy is added to Section 5.
4. The explanation of scattered points in Fig.9 is added to Section A.3.
5. The statement at p.8 and a typo at p.3 are corrected.

All the changes are highlighted in red in the revised manuscript (see the link below).
https://drive.google.com/drive/folders/1vRXKjscBuaWrnKKU6YeBti6Tfbo2yho3