Flat Bands in Magic-Angle Bilayer Photonic Crystals at Small Twists

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Flat Bands in Magic-Angle Bilayer Photonic Crystals at Small Twists

Kaichen Dong, Tiancheng Zhang, Jiachen Li, Qingjun Wang, Fuyi Yang, Yoonsoo Rho, Danqing Wang, Costas P. Grigoropoulos, Junqiao Wu, and Jie Yao

Phys. Rev. Lett. 126, 223601 – Published 2 June 2021

Abstract

The new physics of magic-angle twisted bilayer graphene (TBG) motivated extensive studies of flat bands hosted by moiré superlattices in van der Waals structures, inspiring the investigations into their photonic counterparts with potential applications including Bose-Einstein condensation. However, correlation between photonic flat bands and bilayer photonic moiré systems remains unexplored, impeding further development of moiré photonics. In this work, we formulate a coupled-mode theory for low-angle twisted bilayer honeycomb photonic crystals as a close analogy of TBG, discovering magic-angle photonic flat bands with a non-Anderson-type localization. Moreover, the interlayer separation constitutes a convenient degree of freedom in tuning photonic moiré bands without high pressure. A phase diagram is constructed to correlate the twist angle and separation dependencies to the photonic magic angles. Our findings reveal a salient correspondence between fermionic and bosonic moiré systems and pave the avenue toward novel applications through advanced photonic band or state engineering.

Received 11 January 2021
Accepted 29 April 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.223601

Physics Subject Headings (PhySH)

Nanophotonics Photonic crystals

Superlattices

Coupled mode theory

Atomic, Molecular & Optical

Authors & Affiliations

Kaichen Dong ^1,2,*, Tiancheng Zhang^1,3,*, Jiachen Li ^1,2, Qingjun Wang¹, Fuyi Yang¹, Yoonsoo Rho⁴, Danqing Wang^1,2, Costas P. Grigoropoulos ⁴, Junqiao Wu ^1,2, and Jie Yao ^1,2,†

¹Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA
²Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
³School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, People’s Republic of China
⁴Department of Mechanical Engineering, University of California, Berkeley, California 94720, USA

^*These authors contributed equally to this work.
^†To whom all correspondence should be addressed. yaojie@berkeley.edu

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Vol. 126, Iss. 22 — 4 June 2021

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Images

Figure 1
The TBPC system. (a) Schematic of TBPC with light localized in the AA stacking regions when a photonic magic angle is present (left), along with one representative dispersion of flat moiré bands leading to such localized modes (right). The lattice constant of monolayer honeycomb photonic crystal is $1.2 μ m$ , while each nanodisk is 220 nm high with a diameter of 400 nm. The moiré bands are analytically calculated with a twist angle and interlayer separation of 5.09° and 50 nm, respectively. Note that the band flattening effect occurs in the 2nd and 3rd bands, which are around the 0 meV energy shift. (b) Comparison between the theoretical models for TBG and TBPC. Similar to TBG, the disks in TBPC fall into two categories: disk $a$ and disk $b$ . However, the coupled modes in TBPC are subject to non-negligible optical losses. The lattice constants for graphene and honeycomb photonic crystal are both denoted as $d$ [34, 35]. The reference frame is also illustrated where the $z$ axis is perpendicular to photonic crystal planes.
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Figure 2
Intersite coupling features in TBPC. (a) Top view of TBPC in the real space showing moiré patterns due to a twist (left) and the mini-Brillouin zone hosted by the moiré superlattices (right). (b) TBPC intersite coupling in the reciprocal space, where blue solid dots and green circles stand for the different modes with specific wave vectors in the photonic crystal layer No. 1 and layer No. 2, respectively. The red hexagon denotes the mini-Brillouin zone. (c) The numerically solved TE mode in a nanodisk (left) and the double-degenerated states at the Dirac point in a monolayer honeycomb photonic crystal. The disk positions are indicated by dashed circles. The $| E |$ fields are normalized separately in each panel. (d) The corresponding Dirac-point feature in the photonic band structure along the $M - K - Γ$ direction.
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Figure 3
Photonic moiré band structures. (a)–(d) Energy dispersions and density of states (DOS) for an interlayer separation of 80 nm and twist angles of 2.8°, 3.6°, 4.6°, and 9°, respectively. The energy is referenced to the Dirac-point energy. Note that when the twist angle equals 3.6° and 4.6°, photonic flat bands appear and are highlighted in red. (e) Local bandwidth of the two photonic bands closest to the Dirac-point energy as functions of the twist angle with an interlayer separation of 80 nm. The bandwidth reaches minimum at the photonic magic angles (3.6° and 4.6°).
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Figure 4
Phase diagram of photonic magic angles. (a) The phase diagram showing the normalized local bandwidth with varying twist angles and interlayer separation. (b) Comparison between the local bandwidth, and the integrated $| E |$ in the AA region calculated by numerical simulation, which are normalized by the minimum local bandwidth and the maximum integrated $| E |$ , respectively. The dashed line is a guide to the eye. (c) Evolution of $| E |$ profile in the AA region with different interlayer separation and a twist angle of 5.09°. (d) Numerically calculated real-space $| E |$ profile when the twist angle and interlayer separation are 5.09° and 50 nm, respectively.
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