Symmetry Of Quasicrystals Research Articles

Fractal surface states in three-dimensional topological quasicrystals

We study topological states of matter in quasicrystals, which do not rely on crystalline order. In the absence of a band-structure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a topological insulator. It relies on a combination of noncrystallographic rotational symmetry of quasicrystals and electronic orbital space symmetry, which is the quasicrystalline counterpart of the topological crystalline insulator. The resulting topological state obeys a nontrivial twisted bulk-boundary correspondence and lacks a good metallic surface. The topological surface states, localized on the top and bottom planes respecting the quasicrystalline symmetry, exhibit a different kind of multifractality with probability density concentrated mostly on high-symmetry patches. They form a near-degenerate manifold of immobile states whose number scales proportionally to the macroscopic sample size. This can present an opportunity for a platform for topological surface physics distinct from the crystalline counterpart. Published by the American Physical Society 2024

Unveiling multipole physics and frustration of icosahedral magnetic quasicrystals

Multipolar physics and their hidden orders have been widely discussed in the context of heavy fermions and frustrated magnets. However, despite extensive research, there are few examples of purely multipolar systems in the absence of magnetic dipoles. Here, we show the magnetic behavior of an icosahedral quasicrystal is generally described by multipoles, and in a specific case by pure magnetic octupoles, resulting from the interplay of spin-orbit coupling and crystal field splitting. Importantly, we emphasize that non-crystallographic symmetries of quasicrystals result in multipolar degrees of freedom, in contrast to the conventional crystals. We first classify the characteristics of multipoles and derive the effective spin Hamiltonian. We then explore how frustration and quantum fluctuations induce entangled quantum phases. Our study presents the magnetic icosahedral quasicrystal as a platform for investigating the exotic multipolar physics.

Topological states in quasicrystals

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various types of crystalline topological states to 2D quasicrystals, where real-space expressions of corresponding topological invariants are introduced due to the lack of translational symmetry in quasicrystals. Finally, since quasicrystals possess forbidden symmetries in crystals such as five-fold and eight-fold rotation, we provide an overview of unique quasicrystalline symmetry-protected topological states without crystalline counterpart.

Quasicrystal symmetry in single-crystalline hematite nano octahedron grown from FeOCl nanoparticles

Here, we report the existence of both 2-fold and 3-fold icosahedral quasicrystal symmetries along with the signature of Moiré Fringes within the single crystal hematite nano octahedron. It is grown from the FeOCl (Iron oxychloride) based nucleation centre, obtained by the thermal decomposition of FeCl3.6H2O in the presence of oleic acid and Urea following a La Mer type crystal growth over time.

Quasicrystalline electronic states in twisted bilayers and the effects of interlayer and sublattice symmetries

We study the electronic structure of quasicrystals composed of incommensurate stacks of atomic layers. We consider two systems: a pair of square lattices with a relative twist angle of $\theta=45^\circ$ and a pair of hexagonal lattices with a relative twist angle of $\theta=30^\circ$, with various interlayer interaction strengths. This constitutes every two-dimensional bilayer quasicrystal system. We investigate the resonant coupling governing the quasicrystalline order in each quasicrystal symmetry, and calculate the quasi-band dispersion. The resonant interaction emerges in bilayer quasicrystals if all the dominant interlayer interactions occur between the atomic orbitals that have the same magnetic quantum number. Thus, not only the quasicrystal composed of the widely studied graphene, but also those composed of transition metal dichalcogenides will exhibit the quasicrystalline states. We find that some quasicrystalline states, which are usually obscured by decoupled monolayer states, are more prominent, i.e., "exposed", in the systems with strong interlayer interaction. We also show that we can switch the states between quasicrystalline configuration and its layer components, by turning on and off the interlayer symmetry.

Approximant-based orientation determination of quasicrystals using electron backscatter diffraction

Orientation mapping of quasicrystalline materials is demonstrated using crystalline approximant structures in the technique of electron backscatter diffraction (EBSD). The approximant-based orientations are symmetrised according to the rotational point group of the quasicrystal, including the visualization of orientation maps using proper colour keys for quasicrystal symmetries. Alternatively, approximant-based orientation data can also be treated using pseudosymmetry post-processing options in the EBSD system software, which enables basic grain size estimations. Approximant-based orientation analyses are demonstrated for icosahedral and decagonal quasicrystals.

Symmetry and Asymmetry in Quasicrystals or Amorphous Materials

Quasicrystals (QCs) are long-range ordered materials with a symmetry incompatible with translation invariance. Accordingly, QCs exhibit high-quality diffraction patterns containing a collection of discrete Bragg reflections. Notwithstanding this, it is still common to read in the recent literature that these materials occupy an intermediate position between amorphous materials and periodic crystals. This misleading terminology can be understood as probably arising from the use of models and notions borrowed from the amorphous solid’s conceptual framework (such us tunneling states, weak interference effects, variable range hopping, or spin glass) in order to explain certain physical properties observed in QCs. On the other hand, the absence of a general, full-fledged theory of quasiperiodic systems certainly makes it difficult to clearly distinguish the features related to short-range order atomic arrangements from those stemming from long-range order correlations.

Interlayer Decoupling in 30° Twisted Bilayer Graphene Quasicrystal.

Stacking order has a strong influence on the coupling between the two layers of twisted bilayer graphene (BLG), which in turn determines its physical properties. Here, we report the investigation of the interlayer coupling of the epitaxially grown single-crystal 30°-twisted BLG on Cu(111) at the atomic scale. The stacking order and morphology of BLG is controlled by a rationally designed two-step growth process, that is, the thermodynamically controlled nucleation and kinetically controlled growth. The crystal structure of the 30°-twisted bilayer graphene (30°-tBLG) is determined to have quasicrystal-like symmetry. The electronic properties and interlayer coupling of the 30°-tBLG are investigated using scanning tunneling microscopy and spectroscopy. The energy-dependent local density of states with in situ electrostatic doping shows that the electronic states in two graphene layers are decoupled near the Dirac point. A linear dispersion originated from the constituent graphene monolayers is discovered with doubled degeneracy. This study contributes to controlled growth of twist-angle-defined BLG and provides insights on the electronic properties and interlayer coupling in this intriguing system.

Three‐Dimensional Generalized Dynamics of Soft‐Matter Quasicrystals

The three‐dimensional generalized dynamics of soft‐matter quasicrystals was investigated, in which the governing equations of the dynamics are derived for observed 12‐fold symmetry quasicrystals and possibly observed 8‐ and 10‐fold symmetry ones in soft matter. The solving methods, possible solutions for some initial‐ and boundary‐value problems of the equations, and possible applications are discussed as well.

Tamm plasmon structure with an optical Fibonacci lattice coating

We have theoretically investigated a Tamm plasmon structure covered by subwavelength grating with quasicrystal symmetry using a finite element method. The possibility to support a number of high-Q hybrid optical eigenstates in such systems was demonstrated. The model structure consists of a GaAs/Al0.95Ga0.05As Bragg reflector covered with a F6 subwavelength Fibonacci silver/PMMA coverage. The considered system with a quasicrystalline grating has interesting properties and can be used as light filtering and emission amplifying components of optoelectronic systems.

Tamm plasmon-based cavity with a quasicrystalline type structuration

We have theoretically investigated Tamm plasmon structure covered by a quasicrystal symmetry subwavelength grating. We show that such structures can support plenty of high-Q hybrid optical eigenstates. The structure consists of a GaAs/Al0.95Ga0.05As Bragg reflector covered with a subwavelength Fibonacci silver/resist sequence. For the F5 type coverage, two different operation regimes controlled by filling factor value were shown. We demonstrate the anti-crossing behaviour of these states spectral position. The considered system can be useful to enhance the emission of broadband light-emitting materials such as organic dyes and small molecules.

Topological Boundary Floppy Modes in Quasicrystals

We study topological mechanics in two-dimensional quasicrystalline parallelogram tilings. Topological mechanics has been studied intensively in periodic lattices in the past a few years, leading to the discovery of topologically protected boundary floppy modes in Maxwell lattices. In this paper we extend this concept to quasicrystalline parallelogram tillings and we use the Penrose tiling as our example to demonstrate how these topological boundary floppy modes arise with a small geometric perturbation to the tiling. The same construction can also be applied to disordered parallelogram tilings to generate topological boundary floppy modes. We prove the existence of these topological boundary floppy modes using a duality theorem which relates floppy modes and states of self stress in parallelogram tilings and fiber networks, which are Maxwell reciprocal diagrams to one another. We find that, due to the unusual rotational symmetry of quasicrystals, the resulting topological polarization can exhibit orientations not allowed in periodic lattices. Our result reveals new physics about the interplay between topological states and quasicrystalline order, and leads to novel designs of quasicrystalline topological mechanical metamaterials.

Properties of defect modes in two-dimensional organic octagonal quasicrystals at low dielectric contrast

We study theoretically the localization and coupling properties of defect modes in octagonal quasi-crystals made of conjugated polymers for the light modulation of organic light-emitting devices (OLED) and organic lasers. Due to the rotational symmetry of quasicrystals and the low refractive-index of organic semiconductors, it is observed that localized modes only occur in the complex defect-cavity structures. Unlike conventional photonic crystals with high-index, some unusual characteristics of defect modes coupling in quasicrystals under the low-index contrast are revealed and analyzed, such as the loss of defect modes, composite resonant cavity, and phase-opposition of modal tails. These results will allow us to better utilize the light control in organic photonic crystal microcavity lasers, especially in the localization region, and further to apply in OLED and VCSEL (Vertical Cavity Surface Emitting Laser) fields.

B 7 as a supergroup of crystal and quasicrystal symmetries

In sharp contrast to the generation of a finite group that includes all the 14 types of Bravais lattices as its subgroups [Hosoya (2000). Acta Cryst. A56, 259-263; Hosoya (2002). Acta Cryst. A58, 208], it was proved that a signed permutation group Bk may be interpreted as the supergroup of both crystal and quasicrystal symmetries. Minimal dimension k = 6 is adequate for lattices referred to their three non-coplanar shortest vectors, or for symmetry groups of most quasicrystal types. If one prefers complete, well defined semi-reduced lattice descriptions or needs a dodecagonal group, the B7 supergroup is necessary. All considered matrix groups correspond to isometric transformations in extended k-bases and may be easily derived from B7 and projected onto three-dimensional crystallographic space. Three models of extended bases are proposed: semi-reduced, cyclic and axial. In all cases additional basis vectors are strictly (functionally) related to three original basis vectors.

Origin of 18-fold quasicrystal

We investigate the origin of the 18-fold symmetry in quasicrystals. We perform Monte Carlo simulations of 2D disks interacting with the hard-core/square-shoulder potential, and we analyze the bond-orientational order and the local vertex configurations. We find that the so-called special clusters of particles play a crucial role in the mixing of the three sets of bond directions 20° apart, producing the 18-fold diffraction pattern.

Phonon–phason dynamics and hydrodynamics of fivefold and tenfold symmetry quasicrystals

In view of the same model of fivefold and tenfold symmetry quasicrystals, with the same boundary and the same initial conditions, we have obtained a lot of results by phonon–phason dynamics and hydrodynamics and have performed a detailed comparative analysis. The quantitative results on mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, and viscosity stresses and their time–space variations help us to understand the motion of solid quasicrystals. The analysis for octagonal and dodecagonal quasicrystals can be easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. These results reveal that the phonon field is dominated; moreover, the coupling between phonon and phason fields is important for the studied dynamic process, and these two elementary excitations are the main figures in the dynamic process. In contrast, the effects of the compressibility and viscosity are very weak, and they make almost no contribution to the dynamic process. For this dynamic process, the hydrodynamics and phonon–phason dynamics are equivalent, and the hydrodynamics can be simplified by phonon–phason dynamics for solid quasicrystals.

Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals

This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.

Tiling approach for the description of the sevenfold symmetry in quasicrystals

An example of substitution rules for the construction of heptagonal rhombic tilings is proposed. Rigorous inflation/deflation rules make it possible to expand the tiling up to infinity without additional ad hoc rearrangements. The derived tilings are self-similar and consist of characteristic patterns with seven-pointed stars surrounded by similar seven-pointed stars.

Watching Quasicrystals Grow

The initial discovery in 1984 [1] that certain metallic alloys can possess rotational symmetries forbidden in crystals, such as the symmetry of the regular icosahedron, raised a host of questions about these so-called quasicrystalline materials, or quasicrystals for short. Can the forbidden symmetries of quasicrystals be reconciled with their ability to form x-ray diffraction patterns with sharp spots? Can quasicrystals form faceted polyhedral shapes? Are they thermodynamically stable? These questions and many more have largely been answered over the subsequent decades, and in 2011 Daniel Shechtman was awarded the Nobel Prize in chemistry for the discovery of quasicrystals [2]. However, because quasicrystals have long-range but nonperiodic order, one of the most perplexing questions about them continues to be, “How do quasicrystals grow?” Theoretical models and computer simulations have made important advances on this front [3–6], but experimental answers to this question have been elusive. Using in situ highresolution transmission electron microscopy (HRTEM), Keiichi Edagawa and colleagues [7] at the University of Tokyo and Tohoku University have now successfully observed—at the atomic scale—quasicrystals in the act of growing. For centuries, crystals have been understood, indeed defined, to be composed of a single building block, or “unit cell,” that is copied and stacked together to form the entire three-dimensional crystal structure. The unit cell has been the foundational concept of crystallography from which models and theoretical frameworks have been developed to understand crystals and all of their properties. In particular, our comprehension of the ability of crystals to diffract x rays has—since the pioneering work of Walter Friedrich, Paul Knipping, and Max von Laue in 1912 [8]—been dependent on the long-range, periodic translational symmetry generated by the stacking of unit cells. The structures of quasicrystals, on the other hand, can be modeled based on the famous Penrose tilings—nonperiodic arrangements that have not just one but two unit cells. But there is a dilemma in our understanding of quasicrystal formation. Crystals are built from the repetition of a single unit cell, so only local interactions between atoms are necessary to understand how they might grow. The long-range but nonperiodic order of perfect quasicrystals seems, however, to be significantly more difficult to build, tile-by-tile, using only local interactions. Consider, for example, the tile attachment in a quasiperiodic direction in the Penrose tiling (Fig. 1). Along the quasicrystal growth direction, the ideal tiling has a sequence of wide (W) and narrow (N) rows of tiles, where W and N are arranged in the predictable but nonperiodic fashion of a so-called Fibonacci sequence. To create a Fibonacci sequence, start with the short sequence WN and generate a new sequence following the replacement rules W→WN and N→W, yielding WNW. Subsequent application of these replacement rules to a prior sequence systematically generates a still longer Fibonacci sequence, such as WNWWN and WNWWNWNW (Fig. 1 illustrates part of this latter sequence). But what mechanism ensures that new layers of tiles attach in this specific way during growth? Therein lies the dilemma.

Symmetry of icosahedral quasicrystals

The concept of infinitely fragmented fractal tiling is proposed for the description of the structure and symmetry of quasicrystals. Fractal tilings may serve as unique “parent” structures for the corresponding local isomorphism class. The generating symmetry elements and some special features of the resulting symmetry groups of the fractal tilings are analyzed. Simple inflation/deflation rules for icosahedral quasicrystals are proposed, and natural local matching rules are derived.