Photonic band structure calculations of two-dimensional Archimedean tiling patterns

Ueda, Kazuo; Dotera, Tomonari; Gemma, Tohru

doi:10.1103/physrevb.75.195122

Cited by 74 publications

(71 citation statements)

References 48 publications

(56 reference statements)

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“…In addition, the family of semiregular Archimedean tilings features intriguing characteristics. They may represent geometrically frustrated magnets (7) or provide novel routes for constructing photonic crystals (8). However, with the exception of the frequently realized trihexagonal tiling (also known as the Kagomé lattice) (9)(10)(11)(12)(13)(14)(15), the other semiregular Archimedean tiling patterns remain largely unexplored.…”

mentioning

confidence: 99%

Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly

Écija

Urgel

Papageorgiou

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

127

109

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The tessellation of the Euclidean plane by regular polygons has been contemplated since ancient times and presents intriguing aspects embracing mathematics, art, and crystallography. Significant efforts were devoted to engineer specific 2D interfacial tessellations at the molecular level, but periodic patterns with distinct five-vertex motifs remained elusive. Here, we report a direct scanning tunneling microscopy investigation on the ceriumdirected assembly of linear polyphenyl molecular linkers with terminal carbonitrile groups on a smooth Ag(111) noble-metal surface. We demonstrate the spontaneous formation of fivefold Celigand coordination motifs, which are planar and flexible, such that vertices connecting simultaneously trigonal and square polygons can be expressed. By tuning the concentration and the stoichiometric ratio of rare-earth metal centers to ligands, a hierarchic assembly with dodecameric units and a surface-confined metalorganic coordination network yielding the semiregular Archimedean snub square tiling could be fabricated.T he tiling of surfaces is relevant for pure art (1), mathematics (2, 3), material physics (4), and molecular science (5). Johannes Kepler's pertaining, rigorous analysis revealed four centuries ago that in the Euclidean plane 11 tessellations based on symmetric polygonal units exist (6): three consist of a specific polygon (so-called regular tilings with squares, triangles, or hexagons, respectively), whereas eight require the combination of two or more different polygons (named semiregular or Archimedean tilings from triangles, squares, hexagons, octagons, and dodecagons).Manifestations of regular tessellations at the atomic and molecular level are ubiquitous, including crystalline planes and surfaces of elemental or molecular crystals, and honeycomb structures encountered, e.g., for graphene sheets, strain relief and supramolecular lattices. In addition, the family of semiregular Archimedean tilings features intriguing characteristics. They may represent geometrically frustrated magnets (7) or provide novel routes for constructing photonic crystals (8). However, with the exception of the frequently realized trihexagonal tiling (also known as the Kagomé lattice) (9-15), the other semiregular Archimedean tiling patterns remain largely unexplored.Three of the semiregular Archimedean tilings correspond to five-vertex configurations (Fig. 1 A-C): the snub hexagonal tiling (four triangles and one hexagon at each vertex, labeled 3.3.3.3.6), the elongated triangular tiling (three triangles and two squares join at each vertex in a 3.3.3.4.4 sequence), and the snub square tiling (three triangles and two squares at each vertex; labeled 3.3.4.3.4). They have been identified in bulk materials, such as layered crystalline structures of complex metallic alloys (4, 16-18), supramolecular dendritic liquids (19), liquid crystals (20), special star-branched polymers (21, 22), and binary nanoparticle superlattices (23). Moreover, recent experiments with colloids at a quasicrystalline substra...

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mentioning

confidence: 99%

Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly

Écija

Urgel

Papageorgiou

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

127

109

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“…In this case, the vertical cross sections to the cylindrical axes must have the regular geometrical patterns called Archimedean tiling patterns consisting of regular polygons and there exist only eleven types as illustrated in Fig. 11 [38]. Every vertex in one type of the tilings is under the same environment and a tiling type is denoted by a set of integers standing for polygons, which meet consecutively on each vertex.…”

Section: Abc Miktoarm Star Terpolymersmentioning

confidence: 99%

Block Copolymers and Miktoarm Star-Branched Polymers

Hasegawa

2015

Anionic Polymerization

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Block copolymers self-assemble to form extremely regular nanoscale microdomain structures with a variety of morphologies. The microdomain morphology depends on the chain architectures, number of components, copolymer compositions, segregation power between the components as well as the ordering processes. In this chapter how these factors affect the microdomain morphology is mentioned. (Fig. 1a). On the other hand, a variety of polymer architectures can be produced by connecting the chain ends of three or more polymer chains for miktoarm starbranched polymers (Fig. 1b). KeywordsWhen the segregation power between the block chains of different kinds is strong enough for block copolymers and miktoarm star-branched polymers, they segregate from each other to form a phase-separated structure. However, because of the connectivity between the block chains, large domain structures typically of the order of micron sizes observed in polymer blends cannot grow into block copolymers and miktoarm star-branched polymers. Instead, they form regular periodic domain structures of 10-100 nm size with various morphologies via self-assembly.

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“…Thus, some of the wave-guided modes (trapped modes lying below the light line) will be shifted to the diffracted modes lying above the light line and escape from the device [2]. Photonic quasi-crystals (PhQs) are similar to PhCs but rely on a quasi-crystal arrangement of scattering objects [5,[9][10][11]. PhQs possess high symmetry orders not achievable in nature and offer many desirable features, such as isotropic PBGs, operation in low refractive index materials, diffraction not arising from the nearest neighbor dielectric rod interactions, and flat dispersion bands.…”

Section: Introductionmentioning

confidence: 99%

“…However, it is difficult to numerically establish the dispersion properties of most PhQs. Archimedean lattices are a category of PhQs, consisting of regular convex polygons which are not necessarily identical and can fill the whole plane without gaps [11]. There are 11 Archimedean tilings, including the familiar traditional Bravais lattices: square, triangular, and honeycomb structures.…”

Section: Introductionmentioning

confidence: 99%

Design and Modeling for Enhancement of Light Extraction in Light-Emitting Diodes With Archimedean Lattice Photonic Crystals

Chen¹,

Yeh²,

Chen³

et al. 2009

PIER B

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Abstract-Light extraction efficiency of light-emitting diodes (LEDs) based on various photonic crystal (PhC) structures is investigated in this study. By using the plane wave method and the finite element method, the influence of several factors on the enhancement of light extraction is discussed, including lattice type, the density of states from a photonic band diagram, the ratio of cylinder radius and lattice constant, and the thickness of a PHC pattern. Some rules are given for the practical implementation of an optimized PhC-based LED with higher light extraction efficiency. In the simulation results, the maximum enhancement of the extraction efficiency could be by a factor of approximately 2.8 for the optimized Archimedean tiling pattern in the LED device emitting at the center wavelength of 530 nm. However, with the use of optimized PhC structures, the square and triangular lattices reveal enhancements of ∼ 1.7 and ∼ 1.5, respectively. The extraction efficiency of the Archimedean PhC LED is much greater than that of the regular lattice PhC LED.

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Photonic band structure calculations of two-dimensional Archimedean tiling patterns

Cited by 74 publications

References 48 publications

Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly

Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly

Block Copolymers and Miktoarm Star-Branched Polymers

Design and Modeling for Enhancement of Light Extraction in Light-Emitting Diodes With Archimedean Lattice Photonic Crystals

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