Kwansei Gakuin University
Report of Research Outcome
2024/03/15

To President
Department： Science and Technology
Position： Postdoctoral fellow
Name： PHAN THANH HUYEN
I report the outcome of the research as follows.

Name of theFund/Program□Sabbatical leave with grant□Sabbatical leave with no grant□KGU Joint Research□Individual Special Research⊠Postdoctoral fellow※ Please report by designated form as for “International ResearchCollaboration”.

ResearchTheme

Research
Site/Venue

Research
periodTheoretical Studies for Topological Design of Photonic Crystals

Kwansei Gakuin University – Graduate School of Science and Technology

2023/10/01～2024/03/31 （ 6 months）◆ Summary of the research outcome （ ap p ro x. 2,500 w o rd s ）Please write down the outcomes in detail regarding the research theme above.
With the development of information and telecommunication technology, the techniqueof manipulating and guiding light becomes tremendously attractive. Photonic crystals(PhCs), which is the optical analog of conventional crystals, is the potential structure forthis technique. There are many interesting phenomena of PhCs which can be applied for
devices such as integrated optical circuit and optical switches. However, the energy-loss
due to fabrication and intrinsic properties restrict the further development of photonicdevices. To deal with this problem, the concept of “topology”, which originates frommathematics, was applied to photonic systems. This concept can help distinguish geometry
lattices by some invariant quantities during continuous deformation. Topological photonics
can be understood as the cross field between PhCs and topological physics which help
discover the novel phenomena of optical effects.
The purposes of our research is to theoretically design and find out new topologicalcharacteristics of PhC systems by using numerical calculations. Designing a topological
PhC usually requires breaking the discrete particle symmetry of the crystals, such as
time-reversal symmetry, which is not easy to achieve. Another way is to mimic the existing1

structure of conventional electronic crystals such as graphene, diamond cubic lattice, …
Following the successful application of graphene and other nanoelectronics materials,
the searching for new carbon materials becomes essentials. Recently, Fan, et al, has
successfully synthesized the Biphenylene network (BPN). Then, A. Bafekry et al. has
performed the detailed of density functional theory-based first-principles calculations to
deeply study the electronic and optical properties of this material. Inspired from the initialstudies of the BPN, we have studied the electronic and topological properties of the BPN
by using tight binding model [1]. We also demonstrate that the topological phase transition
occurs when tuning the inter- and intracellular electron hoppings. The topological edge
states are induced by the non-zero Zak phase as shown in Fig. 1(b). Furthermore, we
demonstrate that the exact number of edge states aligns with the number of Wannierorbital at the edge boundary. The similar argument can be used to ilustrate the existence of
topological corner states in the system (Fig. 1(c) and (d)) where uncoupled Wannierorbitals are located at the corners.

Fig. 1. ( a ) T h e l a t t i c e s t r u c t u r e o f t h e B P N . T h e s h a d e d r e g i o n i s t h e u n i t c e l l , c o n t a i n i n g s i x e q u i v a l e n ts u b l a t t i c e s o f c a r b o n a t o m s ( A t o F ) . 𝒂! = (𝑎, 0) a n d 𝒂" = (0, 𝑎 # ) a r e t w o p r i m i t i v e v e c t o r s . T h e i n t r a c e l l u l a r(intercellular) hopping denoted by red (black) lines. (b) Zigzag nanoribbon edge states for different ratio of𝛾 $ /𝛾. T h e e m e r g e n c e o f t o p o l o g i c a l e d g e s t a t e s i s c o n s i s t e n t w i t h t h e Z a k p h a s e c a l c u l a t i o n . ( c ) E n e r g ydispersion for the BPN nanoflake, which shows the isolated corner states in the band gap. (d) Wavefunctionsof the topological corner states.

Although it is very hard to tune the electron hoppings energies in the actual BPN
materials, we can design a system having the topological properties of the BPN based on
photonic crystals (PhCs). In our work, we also introduce a photonic crystal (PhC) structurefollowing the BPN structure, so-called the BPN PhC [2] as shown in Fig. 2(a). We use both
finite element and finite difference methods to study. This BPN PhC shows a complete
band gap when 𝑑 = 1.3𝑎! /3 and 𝑟" = 0.2𝑑. By examining different ribbon structures, we
obtained the topological edge states, which is localized at the zigzag and chiral edges.
These edge states are induced by the non-zero Zak phase in the corresponding directions.
In Fig. 2(b), the upper panel is the schematic of a ribbon with chiral edges, the middle
panel present the Zak phase for the first three bands, the lower panel is the frequency
dispersion of the ribbon structure. The topological edge states are observed in the2complete band gap (blue line), which is consistent with the Zak phase calculation.

Fig. 2. (a) The left panel is the schematic of the BPN PhC. The red rectangle indicates the primitive unit cell,w h i c h c o n t a i n s i x e q u i v a l e n t d i e l e c t r i c r o d s , c o l o r e d b y c y a n . T h e r a d i u s o f e a c h r o d i s 𝑟% , t h e d i s t a n c eb e t w e e n e a c h r o d a n d t h e c e n t e r o f t h e u n i t c e l l i s d , ) . 𝒂! = (𝑎, 0) a n d 𝒂" = (0, 𝑎 # ) a r e t w o p r i m i t i v e v e c t o r s .
T h e r i g h t p a n e l i s t h e p h o t o n i c b a n d s t r u c t u r e f o r 𝑑 = 1.3𝑎 # /3, 𝑟% = 0.2𝑑. ( b ) A s c h e m a t i c o f a r i b b o n w i t h c h i r a ledges (upper panel), the corresponding Zak phase (middle panel) and the frequency dispersion (lower panel)of the ribbon. The blue line indicates the topological edge states. (c) A BPN corner structure, which contains
chiral edges, the frequency dispersion of the corner structure, which shows the isolated corner states marked
by red dots in the complete band gap. The field distribution for the corner states.

Similar to the studies of the BPN, here we also examine a flake structure, which contains
chiral edges and four corners as shown in Fig. 2(c). The frequency dispersion show four
isolated states in the band gap, labeled by re dots. These states are topological cornerstates, where the electromagnetic (EM) waves are localized at the corner andexponentially decay. Compared with the graphene-like PhC, the topological states in
BPN-like PhC can be found without breaking the symmetry of the crystals. Our results
suggest a possible way to design in-gap topological waveguide and topologicalconfinement of EM waves.
We studied the electronic and photonic BPN parallelly. The results of our work on theBPN and its photonic counterpart are quite consistent and play an important role in thestudies of both the BPN and the BPN PhC.
Beside the studies of the 2D systems, we also extend pur study to 3D PhC systems. 3D
systems can provide various topological phenomena, as the third dimension expands theopportunities for the structural design of PhC or metamaterials. The topological states in3D systems have been mainly studied in acoustic systems, there are so few report on thePhC with complete band gap. In our work, we numerically and theoretically study thetopological properties of 3D simple cubic PhC [3] as shown in Fig. 3(a). This PhC structure
show a complete band gap. We focus on examining the topological properties of thesystems in this band gap. At the beginning, we report a numerical method for calculating
the Zak phase of 3D PhC. The topological interface states are observed in the interface
structure due to the finite different in the Zak phase between two types of unit cell. Owing3

to the inversion symmetry protection, the topological hinge states are found when the redunit cells are embedded inside the blue unit cells. Our research finding can be applied tothe 3D control of microwave propagation for communication and is an important step
towards the realization of robust 3D photonic circuits for infrared light.

Fig. 3. (a) A schematic of two types of unit cells in simple cubic lattice and their photonic band structure,
which show a complete band gap. (b) The interface structure between two types of unit cell (upper panel). The
frequency dispersion is shown in the lower panel. In the complete band gap, two topological interface states
are numerically observed, where the EM wave is highly localized at the interface. (c) The supercell where thered unit cells are embedded at the center and surrounded by the blue unit cells. This supercell contains 4
equivalent hinges. The bulk-interface-hinge correspondence is shown in the left panel, where bulk, interface
and hingh states are colored in gray, blue and red.

In conclusion, in the last 6 months, we mainly focus on studying the topologicalproperties of photonic crystals. Since the inversion symmetry is preserved in our systems,
we use the Zak phase as a topological invariant to distinguish topological properties. In thenext period, we may extend the studies to the systems without inversion symmetry or in the
non-Hermitian systems.
Published papers:[1] K. Koizumi, H. T. Phan, K. Nishigomi, K. Wakabayashi, Phys. Rev. B 109 035431 (2024)[2] H. T. Phan, K. Koizumi, K. Wakabayashi, Opt. Express 32 2223-2234 (2024)[3] S. Takahashi, Y. Ashida, H. T. Phan, et al., Phys. Rev. B 109 125304 (2024)
Academic conferences:[4] Higher Order Topological States in 3D Woodpile Photonic Crystal, H. T. Phan, et al.,JST CREST Topology Joint Seminar, 9 Nov. 2023 (Komaba II Campus, University of Tokyo)[5] H. T. Phan, K. Wakabayashi, Skin effect caused by point-gap topology in non-Hermitian
photonic crystal, JSAP Spring Meeting 2024, Mar. 22rd, Tokyo City University.
Deadline： Within two months after finishing the research period.
Sabbatical leave with grant: Submit this report to President with confirmation by the dean ofschool you belong to.
※ Postdoctoral fellow is required to submit this report with confirmation by the dean of graduate
school before the end of employment period.
Where to submit： Organization for Research and Development and Outreach (NUC)4

◆ We put this report on the web of KGU. If there is any problem about it because of difficulties on
your research, please let us know.

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