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Vortex states in a PbTiO$_3$ ferroelectric cylinder
by Svitlana Kondovych, Maksim Pavlenko, Yurii Tikhonov, Anna Razumnaya, Igor Lukyanchuk
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Submission summary
| Authors (as registered SciPost users): | Svitlana Kondovych |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2112.10129v2 (pdf) |
| Date submitted: | 2022-08-19 14:19 |
| Submitted by: | Kondovych, Svitlana |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational, Phenomenological |
Abstract
The past decade's discovery of topological excitations in nanoscale ferroelectrics has turned the prevailing view that the polar ground state in these materials is uniform. However, the systematic understanding of the topological polar structures in ferroelectrics is still on track. Here we study stable vortex-like textures of polarization in the nanocylinders of ferroelectric PbTiO$_3$, arising due to the competition of the elastic and electrostatic interactions. Using the phase-field numerical modeling and analytical calculations, we show that the orientation of the vortex core with respect to the cylinder axis is tuned by the geometrical parameters and temperature of the system.
Author comments upon resubmission
We acknowledge the Editor and Referees for the careful reading and review of our work. We thank the Reviewers that they found our work interesting, recommended it for publication in SciPost Physics, and provided constructive comments for improving the manuscript. We have accounted for these insightful comments and addressed the Referees’ concerns point-by-point below.
Answers to the Report 1 on 2022-7-7
- Reviewer: Equation (5) gives the polarization distribution of an isotropic vortex in the cylindrical nanodot. However, this expression would not fulfill the natural boundary condition of zero polarization derivatives at r=R. Comment on this would be useful in the manuscript.
Answer: The solution given by Equation (5) is approximate and corresponds to the long-range asymptotic behaviour of polarization when r≫ξ0. Accordingly, the polarization derivative at r=R scales as (ξ0/R) and thus is approaching zero when ξ0/R ≪ 1. In this limit, it fits the natural boundary conditions and is in agreement with the numerical simulations as seen in Fig. 7a of the resubmitted manuscript. We thank the Reviewer for pointing out this issue and have included the corresponding remark in Appendix A.2, where the solution for the radial polarization distribution is obtained (page 14).
- Reviewer: In numerical simulations, the ferroelectric cylinders are surrounded by non-ferroelectric medium as shown in Fig.5. In the text, the cylinders are described as free-standing particles. I think it would be practical to mention the requirements for this surrounding media needed to consider the cylinders as free-standing, as well as to discuss the possible technological realizations of such free-standing nanostructures.
Answer: In our study, we consider the cylinder being embedded in vacuum with ε=1. For technical reasons of calculations, the stiffness coefficients were considered finite but very small that practically corresponds to the vacuum situation. Therefore our model corresponds to the free-standing nanostructures that are frequently realized in practice; see, for instance, Ref.24.
Answers to the Comment on 2022-05-18
- Reviewer: The authors stated that the basic idea of the polarization vortices formation is sketched in Fig. 1. Under short-circuited boundary conditions, the uniform polarization occurs in the PbTiO3 cylinder, while the vortex states are observed in the PbTiO3 cylinder under open-circuited boundary conditions. However, as reported by Li S. et al. (Appl. Phys. Lett. 111, 052901 (2017)), the symmetry of the electrical boundary conditions played more crucial influences on the formation of flux-closure domains. Thus, additional theoretical results or discussions are suggested to be included in this paper to illuminate the formation and transition of vortex states in ferroelectric cylinders.
Answer: We thank the Reviewer for drawing our attention to this research, which focuses on the important role of the electrodes in formation of ferroelectric domains. Indeed, contact with oxide electrodes may affect the depolarization field screening and lead to changes in the domain structure. However, in our model sketched in Fig.1 we consider ideal metallic electrodes with much smaller screening length compared to oxide electrodes discussed by Li S. et al. These considerations are beyond the scope of our work, which focuses on the vortex state formation in free-standing cylinders, but present an appealing problem for the future. Therefore, we add the corresponding remark to Discussion, Section 4.1, and highlight the mentioned research in References.
- Reviewer: Figure 7 is suggested to be included in the main text, which would facilitate to understand the transition between a-vortex states and c-vortex states displayed in the phase diagram in Fig. 2.
Answer: We have moved Figure 7 from Appendix A.3 to Section 4.4 of the main text as suggested (in the resubmitted version, it becomes Figure 4 on page 9).
- Reviewer: About the title, “in a ferroelectric cylinder” is suggested to be more specific, as in the manuscript, only PbTiO3 cylinder was discussed.
We have changed the title to “Vortex states in a ferroelectric PbTiO$_3$ cylinder” to be more specific as suggested by the Reviewer.
List of changes
1. The title was changed to “Vortex states in a ferroelectric PbTiO$_3$ cylinder”.
2. Section 4.1 was expanded with a discussion on possible types of electrodes for depolarization field screening.
3. Figure 7 was moved from Appendix A.3 to Section 4.4 of the manuscript (Fig 4 on page 9 in the resubmitted version).
4. A comment on polarization boundary conditions was added to Appendix A.2.
5. Reference [28] was added.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022-9-19 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2112.10129v2, delivered 2022-09-19, doi: 10.21468/SciPost.Report.5713
Report
This paper present a complete theoretical study of the topological polarization states in free-standing PbTiO3 cylinders, as a function of their length, radius, and temperature, resulting in a detailed phase diagram showing the transition from c-phase to the formation of a- or c-vortices or the composition of the two, arising from the competition of the elastic and electrostatic interactions. The calculations are based on the free energy density functional, simplified for the analytical calculations or using the phase-field numerical modeling, resulting in comparable results.
The topic is particularly timely, as the interest of the community in exotic polarisation textures and topologies in lead titanate has never been higher, with recent reports of skyrmions, vortices, and bubble domains. The results presented here will lead to a better theoretical understanding of this technologically important material, allowing a more systematic control over its more complex polarisation topologies, and paving the way for many future discoveries.
The paper is well written and accessible to theoreticians as well as experimentalists.
For these reasons, I recommend the paper to be published with only minor changes/suggestions.
1. In the case of a-vortices, how is the orientation of the in-plane vortex-axis determined? Is itartificially aligned with the [100] or [010] axis or can it take any other in-plane direction?
2. 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?
3. 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.
4. 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, therefor one expects an important role played be flexoelectricity.
5. 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 later 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?
6. p.8 "At larger radii, R>14nm..." this statement is valid for h = 6nm.
7. 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.
8. Figure 9b, one observes a clear deviation from the linear scaling for large R - any comment/explanation?
9. Typo p.3: reach -> rich
Strengths
as specified in the previous report
Weaknesses
as specified in the previous report
Report
The authors have addressed all issues and questions from the reports. The paper is of high scientific quality, it is acceptable for publication in its present from
Requested changes
no further requests