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THz HigherOrder Topological Photonics in GeonSi Heterostructures
by Ian Colombo, Pietro Minazzi, Emiliano Bonera, Fabio Pezzoli, and Jacopo Pedrini
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Submission summary
Authors (as registered SciPost users):  Ian Colombo · Jacopo Pedrini · Fabio Pezzoli 
Submission information  

Preprint Link:  scipost_202311_00045v2 (pdf) 
Date submitted:  20240328 19:06 
Submitted by:  Pedrini, Jacopo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We design germaniumbased higherorder topological cavities for terahertz applications by breaking the symmetry of a twodimensional photonic crystal following the SuSchriefferHeeger model. Calculations demonstrate the parity inversion of the electric field in differently deformed unit cells. The interface between domains of opposite topology presents edge and corner modes. The former are chiral, locking light propagation to its helicity. The latter prove that Gebased structures can be used as highorder topological photonic crystals. These findings can accelerate the development of Siphotonic components working in a spectral range of high technological interest.
Author comments upon resubmission
we thank you for providing us with the referee report. We thoroughly answered to each of their questions. We modified the work by following their report and we believe that this allowed us to improve the quality of our work.
Best regards,
Jacopo Pedrini
List of changes
1. We added the supplementary materials in the form of appendices.
2. We added the following paragraph to the main text (starting at line 117):
"The topological invariant in SSHlike 2D photonic crystals like those described in this work can be classified by the Zak phase [40, 56], which is basically the integral of the Berry connection on the Brillouin Zone. In some works [40] the bulk polarization P is discussed instead of the Zak phase φZ, but the two are simply related by φZ = 2πP. The values of the Zak phase form a Z2 index in C4symmetric topological crystals such as the one described in our work and can only take the values 0 or π for each direction for trivial or nontrivial topologies, respectively. [41, 57] It is known from the literature [40, 41] that in structures akin to those described in this work, the Zak phase for the directions (x , y) is (0, 0) for the compressed structure and (π, π) for the expanded structure, meaning that the structures are topologically trivial and nontrivial, respectively."
We also added references [40,41].
3. We corrected a small mistake in Figure 2g and h, where the electric field distribution maps were inverted.
4. We changed the sentence starting at line 136 with: “One of the fingerprints of a topological transition is the socalled bulk, edge correspondence, that is the emergence of spatially confined guided modes at the boundary between two domains with different band topology”.
5. We added reference [1] to the list of references found at line 138, that is now "[1, 5, 7, 58, 59]"
6. We added a section in the Appendices describing the role of the {111} facets and added at line 63 "The {111} facets are subwavelength and their role on determining the photonic bandstructure of the crystal is negligible, as shown in Figure 5 in Appendix A."
7. We added a section in the Appendices describing the computational methods in detail, and we added at line 75: "Further details on the simulation methods are reported in Appendix B."
8. We corrected some typos as suggested by the reviewer.
Current status:
Reports on this Submission
Anonymous Report 2 on 202457 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202311_00045v2, delivered 20240506, doi: 10.21468/SciPost.Report.9001
Report
I thank the authors for their detailed responses to my comments. However, I am left with the impression that, conceptually, the model and results closely resemble those found in the reference [Xie et al., Phys. Rev. Lett], which shares similar symmetries, topologies, and results including the same type of phase transition. The theoretical added value of the modifications remains unclear to me. While I acknowledge that using different materials and designs can indeed lead to new applications in industry, this suggests that the work may be more suitable for publication in specialized engineering journals.
Two additional points:
1. Despite the correction in Figure 1, my initial concern from the previous report persists. It is unclear how a trivial phase and a nontrivial one can be simply related by lattice translation. This criticism applies to reference 40 as well. Nevertheless, it would be beneficial to clarify this point.
2. As clear from Appendix A, the role of the facet (111) is negligible. It's not clear what is the advantage of including it in the design. One can imagine a large number of small modifications that will turn out to be negligible. I believe that the simplest functional forms (like a perfect square) are preferred at least from a fabrication perspective. Of course, one can ask the question regarding the order of magnitude of perturbations such that the results are not impacted. This question might have a practical relevance in terms of how precise the fabrication process has to be. However, the purpose of the inclusion facet (111) doesn't seem to me to provide an answer to the question.
Recommendation
Accept in alternative Journal (see Report)
Anonymous Report 1 on 202453 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202311_00045v2, delivered 20240503, doi: 10.21468/SciPost.Report.8952
Report
This is a theoretical investigation, building on the experimental investigation of Phys. Rev. Applied 16, 064024 (2021), Ref. [37]. The calculations are carried out with the COMSOL Multiphysics applications suite version 6.1.
I have to confess that this investigation is peripheral to my own interests and that I have not reviewed the first version of the manuscript. However, I have the impression that this is more on the level of routine work and thus more suitable for publication in SciPost Physics Core. After all, the physics is singleparticle physics and the computations are carried out with a commercial program package.
There is one item concerning content that has been mentioned before (point 1 in Anonymous Report 1 on 2024225), but in my opinion has not been properly addressed yet: The "Compressed" and "Expanded" unit cells of Figs. 2a) and c) are obviously equivalent in the bulk, and indeed the band structures of Figs. 2d) and f) seem to be identical. Accordingly, their symmetry groups must be isomorphic, i.e., only representations can differ. Of course, if one cuts the unit cells in different manners, the edge states can be different. Nevertheless, I find the comments on the bulk properties in Anonymous Report 1 on 2024225 clearer than the corresponding discussion on page 5 of the manuscript. I thus recommend revisiting the discussion on page 5; first focusing on bulk properties and only in a second step introducing boundary states and topological properties might help.
Beyond this, there are some details that I recommend to address, see "Requested changes".
Requested changes
1 The \cite commands after the punctuation marks are sometimes confusing; at least they look strange. While I admit that there seem to be no specific formatting instructions in this respect, I recommend moving the \cite commands before the punctuation marks.
2 The inset of Fig. 1(c) is too small to be readable, at least in my printout. Enlarging the entire figure might help, but to play things safe, I recommend to also revisit the inset.
3 Revisit discussion of "Compressed" and "Expanded" unit cells on page 5, acknowledging that they are equivalent in the bulk.
4 The graphs in Fig. 3b) suggest a small value of $d/a$, but I was not able to find the exact value. I recommend providing this information in the caption of Fig. 3.
5 Figure 3c) contains information that is again too small to be easily visible in a printout. SciPost has no limitation on the length of the manuscript. So, why not use the space it takes for a clear figure?
6 Due to the multiple adjectives, I was not able to make sense of the "opportunely spaced phased dipoles" on lines 189  maybe reconsider a clearer formulation.
7 Thanks to a comment on line 65, one knows that Fig. 5 belongs to Appendix A. However, Appendix A does not refer to Fig. 5. I recommend adding at least a short comment also at this place.
8 On lines 229230 there is a reference to Figure S4, but that figure does not seem to exist.
9 I found Fig. 8 a bit confusing since the horizontal axis is reset with each new $k$ point. Maybe a graph with two panels would be clearer: one showing the number of iterations needed and another one the final error, both as a function of $k$.
10 Ref. [16] lacks the article identifier eaat0346 and DOI 10.1126/sciadv.aat0346.
11 Ref. [35]: The "(80. )." after "Science" looks strange to me.
12 Ref. [38] might be better readable without the URL (i.e., DOI should suffice).
13 The DOI for Ref. [43] is 10.1364/OE.27.016088  please add.
14 Ref. [49]: upper/lowercasing in the chemical formulas in the title needs to be corrected, same for subscripts (e.g., "YbF$_3$" etc.).
Recommendation
Accept in alternative Journal (see Report)
Author: Jacopo Pedrini on 20240520 [id 4500]
(in reply to Report 1 on 20240503)We thank the reviewer for their report. Our response is attached.
Author: Jacopo Pedrini on 20240520 [id 4499]
(in reply to Report 2 on 20240507)We thank the reviewer for their comments. Our reply is attached.
Attachment:
Response_reviewer1.pdf