Long-range Quasiperiodic Order Research Articles

Radiative heat transfer mediated by topological phonon polaritons in a family of quasiperiodic nanoparticle chains

Topological phonon polaritons (TPhPs) are localized boundary modes arising from the combination of topological photonics and phonon polaritons in micro/nanostructures made of polar dielectrics, which are capable of mediating long-range radiative heat transfer and infrared sensing in a robust manner. This work shows TPhPs can be sustained in a family of one-dimensional quasiperiodic silicon carbide nanoparticle (NP) chains, and can significantly improve radiative heat transfer for these arrays. This family of 1D quasiperiodic lattices is a continuous interpolation between the two paradigmatic limits, viz., the off-diagonal Aubry-André-Harper (AAH) chain and the Fibonacci chain. For different interpolation parameters τ, the band structures are calculated with respect to the modulation phase ϕ as the wavenumber in a synthetic dimension, where a series of gaps and midgap edge states can be observed. These edge states are shown to be topologically protected by applying the gap-labeling theorem and the principle of bulk-boundary correspondence. By means of many-body radiative heat transfer theory for a group of electric dipoles, it is shown quasiperiodicity and topological property can have an important impact on radiative heat transfer. These results can give useful insights on the interplay among topological order, long-range quasiperiodic order, near-field electromagnetic interactions and radiative heat transfer.

Robust matching rules for real quasicrystals

We consider the problem of extraction and validation of matching rules, directly from the phased diffraction data of a quasicrystal, and propose an algorithmic procedure to produce the rules of the shortest possible range. We have developed a geometric framework to express such rules together with a homological mechanism enforcing the long-range quasiperiodic order. This mechanism tolerates the presence of defects in a robust way.

Thermoelectric transport in poly(G)-poly(C) double chains

Electronic and phononic transport in DNA systems with macroscopic length are studied by means of a real-space renormalization method within the Boltzmann formalism, where the poly(G)-poly(C) base-pair segments arranged following periodic and Fibonacci sequences are comparatively analyzed. The fishbone model and the two-site coarse grain model based on the Born potential including central and non-central interactions are respectively used for the calculation of electrical and lattice thermal conductivities of these DNA systems connected to two reservoirs at their ends. The results show the appearance of gaps in phononic transmittance spectra of segmented poly(G)-poly(C) double chains, which leads to a better thermoelectric figure of merit (ZT) than that of corresponding non-segmented systems. Such ZT can be further improved by introducing a long-range quasiperiodic order, which avoids the thermal transport of numerous low-frequency phonons responsible of the lattice thermal conduction at low temperature. Finally, the influence of reservoirs on ZT is also investigated.

Building 2D quasicrystals from 5-fold symmetric corannulene molecules

The formation of long-range ordered aperiodic molecular films on quasicrystalline substrates is a new challenge that provides an opportunity for further surface functionalization. This aim can be realized through the smart selection of molecular building blocks, based on symmetry-matching between the underlying quasicrystal and individual molecules. It was previously found that the geometric registry between the C60 molecules and the 5- and 10-fold surfaces was key to the growth of quasiperiodic organic layers. However, an attempt to form a quasiperiodic C60 network on i-Ag-In-Yb substrates was unsuccessful, resulting in disordered molecular films. Here we report the growth of 5-fold symmetric corannulene C20H10 molecules on the 5-fold surfaces of i-Ag-In-Yb quasicrystals. Low-energy electron diffraction (LEED) and scanning tunneling microscopy (STM) revealed long-range quasiperiodic order and 5-fold rotational symmetry in self-assembled corannulene films. Recurrent decagonal molecular rings were seen, resulting from the decoration of specific adsorption sites with local pentagonal symmetry by corannulenes, adsorbed with their bowl-openings pointing away from the surface. They were identified as (Ag, In)-containing rhombic triacontahedral (RTH) cluster centers and pentagonal Yb motifs, which cannot be occupied simultaneously due to steric hindrance. It is proposed that symmetry-matching between the molecule and specific substrate sites drives this organization. Alteration of the molecular rim by the introduction of CH substituents appeared to increase molecule mobility on the potential energy surface and facilitate trapping at these specific sites. This finding suggests that rational selection of molecular moiety enables the templated self-assembly of molecules leading to an ordered aperiodic corannulene layer.

Quasiperiodic canonical-cell tiling with pseudo icosahedral symmetry

Icosahedral quasicrystals and their approximants are generally described as packing of icosahedral clusters. Experimental studies show that clusters in various approximants are orderly arranged, such that their centers are located at the nodes (or vertices) of a periodic tiling composed of four basic polyhedra called the canonical cells. This so called canonical-cell geometry is likely to serve as a common framework for modeling how clusters are arranged in approximants, while its applicability seems to extend naturally to icosahedral quasicrystals. To date, however, it has not been proved yet if the canonical cells can tile the space quasiperiodically, though we usually believe that clusters in icosahedral quasicrystals are arranged such that quasiperiodic long-range order as well as icosahedral point symmetry is maintained. In this paper, we report for the first time an iterative geometrical transformation of the canonical cells defining a so-called substitution rule, which we can use to generate a class of quasiperiodic canonical-cell tilings. Every single step of the transformation proceeds as follows: each cell is first enlarged by a magnification ratio of τ3 (τ=goldenmean) and then subdivided into cells of the original size. Here, cells with an identical shape can be subdivided in several distinct manners depending on how their adjacent neighbors are arranged, and sixteen types of cells are identified in terms of unique subdivision. This class of quasiperiodic canonical-cell tilings presents the first realization of three-dimensional quasiperiodic tilings with fractal atomic surfaces. There are four distinct atomic surfaces associated with four sub-modules of the primitive icosahedral module, where a representative of the four submodules corresponds to the Σ=4 coincidence site module of the icosahedral module. It follows that the present quasiperiodic tilings involve a kind of superlattice ordering that manifests itself in satellite peaks in the diffraction patterns. Moreover, we argue that the superlattice ordering causes the overall point symmetry of the tilings to reduce from icosahedral to tetrahedral, although the symmetry reduction is weak and mainly observed in the satellite peaks. The present discovery suggests that the canonical-cell geometry may play an important role in uncovering the mystery of structure and formation of icosahedral quasicrystals.

Spin waves in planar quasicrystal of Penrose tiling

We investigated two-dimensional magnonic structures which are the counterparts of photonic quasicrystals forming Penrose tiling. We considered the slab composed of Ni (or Py) disks embedded in Fe (or Co) matrix. The disks are arranged in quasiperiodic Penrose-like structure. The infinite quasicrystal was approximated by its rectangular section with periodic boundary conditions applied. This approach allowed us to use the plane wave method to find the frequency spectrum of eigenmodes for spin waves and their spatial profiles. The calculated integrated density of states shows more distinctive magnonic gaps for the structure composed of materials of high magnetic contrast (Ni and Fe) and relatively high filling fraction. This proves the impact of quasiperiodic long-range order on the spectrum of spin waves. We also investigated the localization of spin wave eingenmodes resulting from the quasiperiodicity of the structure.

Optical probing of long-range spatial correlation and symmetry in complex biophotonic architectures on transparent insect wings

We experimentally probe the structural organization of complex bio-photonic architecture on transparent insect wings by a simple, non-invasive, real-time optical technique. A stable and reproducible far-field diffraction pattern in transmission was observed using collimated cw and broadband fs laser pulses. A quantitative analysis of the observed diffraction pattern unveiled long-range quasi-periodic order in the arrangement of the microstructures over mm scale. These observations agree well with the Fourier analysis of SEM images of the wing taken at various length scales. We propose a simple quantitative model based on optical diffraction by an array of non overlapping microstructures with minimal disorder which supports our experimental observations. We observed a rotation of the original diffraction profile by scanning the laser beam across the wing sample which gives direct signature of organizational symmetry in microstructure arrangements at various length scales. In addition, we report the first optical detection of reorganization in the photonic architecture on the Drosophila wings by various genetic mutations. These results have potential for the design and development of diffractive optical components for applied photonics and may open up new opportunities in biomimetic device research.

Self-Organized Molecular Films with Long-Range Quasiperiodic Order

Self-organized molecular films with long-range quasiperiodic order have been grown by using the complex potential energy landscape of quasicrystalline surfaces as templates. The long-range order arises from a specific subset of quasilattice sites acting as preferred adsorption sites for the molecules, thus enforcing a quasiperiodic structure in the film. These adsorption sites exhibit a local 5-fold symmetry resulting from the cut by the surface plane through the cluster units identified in the bulk solid. Symmetry matching between the C60 fullerene and the substrate leads to a preferred adsorption configuration of the molecules with a pentagonal face down, a feature unique to quasicrystalline surfaces, enabling efficient chemical bonding at the molecule-substrate interface. This finding offers opportunities to investigate the physical properties of model 2D quasiperiodic systems, as the molecules can be functionalized to yield architectures with tailor-made properties.

A comparative scanning tunneling spectroscopy investigation of the (12110)-surface of decagonal Al–Ni–Co and the (100)-surface of its approximant Y–Al–Ni–Co

We present a low-temperature scanning tunneling spectroscopy study of the local density of states (LDOS) on the (12110)-surface of the decagonal quasicrystal d–Al–Ni–Co and the (100)-surface of its structurally closely related approximant Y–Al–Ni–Co in the range of ±1 eV around the Fermi energy. The LDOS of both surfaces are dominated by a large parabolic pseudogap centered at the Fermi energy, which can be attributed to the Hume–Rothery stabilization mechanism or to an orbital hybridization. Superimposed on this large pseudogap, a spatially varying fine structure is revealed, whose spatial distribution correlates with the structures of both of the surfaces. This fine structure shows narrow peaks and pseudogaps exhibiting typical energy widths between 50 and 150 meV. The spatial extent of these localized states is of the order of the width of the topographic features, which is about 0.45 nm. Our findings show that specific electronic states are localized on equivalent topographic features regardless of the presence of periodic or quasiperiodic long-range order. We interpret this result as an indication that the fine structure in the LDOS is dominated much more by the complex short-range order than by the presence of the quasiperiodic long-range order.

Formation and Properties of Quasicrystals

This article summarizes the general structural features and important properties of quasicrystals, a relatively new class of materials belonging to solid-state substances. Quasicrystals, which have a long-range quasiperiodic order but no three-dimensional translational periodicity, are structurally different from both conventional crystals and glassy solids. Although applications are still highly limited, quasicrystals are attracting significant attention as a structual constituent of mixed-phase materials exhibiting high strength together with certain ductility and can be applied as structural materials.

Highly-perfect decagonal quasicrystalline Al64Cu22Co14 with non-centrosymmetry

The structure of a decagonal quasicrystal of Al64Cu22Co14, which was a single-grain specimen (a few mm in size) grown by slow-cooling from a melt and subsequently annealed at 1073 K, was investigated using electron diffraction and scanning transmission electron microscopy (STEM). Electron diffraction patterns of the Al64Cu22Co14 revealed a large number of sharp reflections that indicate a good long-range quasiperiodic order, and convergent-beam electron diffraction (CBED) analysis identified the space group of the decagonal Al64Cu22Co14 as being non-centrosymmetric . Atomic-resolution Z-contrast STEM imaging showed that the structure can be described based on a fivefold-symmetric decagonal cluster 2 nm across, most of which is found to be located at ideal positions in the pentagonal Penrose tiling with significantly less phason disorders. Therefore, to our knowledge, the present decagonal Al64Cu22Co14 provides one of the best quasiperiodic ordered structures among those belonging to non-centrosymmetric . Further details of atomic configurations within the fivefold-symmetric cluster were also examined using ultrahigh-resolution Z-contrast STEM with an aberration-corrector for the objective lens. This clearly revealed both the Al and Cu/Co sites at resolution close to 1 Å. Atomic arrangements within the cluster are found to be fairly well described based on the fivefold-symmetry cluster that exists in the W-(AlCoNi) approximant crystal, except for the atomistic disorders localized around the cluster centres.

準結晶研究の最近の展開 ３．準結晶のかたち（準結晶の構造）収束電子回折法によって明らかにしたＣｄ系正２０面体準結晶の特異な対称性

Symmetry of icosahedral quasicrystals of Cd-Mg-Yb and Cd-Yb has been investigated by the convergent-beam electron diffraction (CBED) method.The CBED pattern symmetries of the Cd-based quasicrystals are not consistent with any icosahedral point groups, indicating that the structures of the alloys have no icosahedral symmetry. Selected-area electron diffraction patterns of the alloys show regular icosahedral symmetry, indicating that the alloys has a regular icosahedral lattice with a long-range quasiperiodic order. We consider that the structure of the Cd-based quasicrystals is formed by a quasiperiodic arrangement of a non-icosahedral atom-cluster with the same orientation.

Soft quasicrystals–Why are they stable?

In the last two years we have witnessed the exciting experimental discovery of soft matter with nontrivial quasiperiodic long-range order–a new form of matter termed a soft quasicrystal. Two groups have independently discovered such order in soft matter: Zeng et al. in a system of dendrimer liquid crystals; and Takano et al. in a system of ABC star-shaped polymers. These newly discovered soft quasicrystals not only provide exciting platforms for the fundamental study of both quasicrystals and of soft matter, but also hold the promise for new applications based on self-assembled nanomaterials with unique physical properties that take advantage of the quasiperiodicity, such as complete and isotropic photonic band-gap materials. Here we provide a concise review of the emerging field of soft quasicrystals, suggesting that the existence of two natural length-scales, along with three-body interactions, may constitute the underlying source of their stability.

Bose-Einstein condensates in optical quasicrystal lattices

We analyze the physics of Bose-Einstein condensates confined in 2D quasi-periodic optical lattices, which offer an intermediate situation between ordered and disordered systems. First, we analyze the time-of-flight interference pattern that reveals quasi-periodic long-range order. Second, we demonstrate localization effects associated with quasi-disorder as well as quasiperiodic Bloch oscillations associated with the extended nature of the wavefunction of a Bose-Einstein condensate in an optical quasicrystal. In addition, we discuss in detail the crossover between diffusive and localized regimes when the quasi-periodic potential is switched on, as well as the effects of interactions.

Degree of structural perfection of icosahedral quasicrystalline grains investigated by synchrotron X-ray diffractometry and imaging techniques

A study of the structural perfection of icosahedral quasicrystalline grains of various alloys (Al-Pd-Mn, Zn-Mg-RE (RE L rare earth) and Al-Cu-Fe), grown by different slow solidification techniques (Czochralski, Bridgman, flux and annealing) was performed using high-resolution diffraction, including recording rocking curves combined with X-ray topography and phase contrast radiography, at a third-generation synchrotron radiation source (European Synchrotron Radiation Facility, Grenoble, France). For Al-Pd-Mn, additional coherent diffraction and diffuse scattering measurements were also carried out. After evaluating the potentialities of the techniques used, in the light of the criteria defined for crystals, it is shown that the structural perfection of icosahedral quasicrystals is quite comparable with that of metallic crystals but is considerably influenced by either uniform phason strains which can destroy the quasiperiodic long-range order, or by long-wavelength phason fluctuations leading to diffuse scattering. The structural perfection was also found to be extremely variable across the as-grown quasicrystalline grains and to be dependent on the presence and characteristics of inhomogeneities (pores and precipitates) often included in the quasicrystalline matrix. Regarding the grains that we used, it has been impossible to distinguish a clear influence of either the type of alloy or the growth method. It has, however, been noticed that Al-Pd-Mn and Al-Cu-Fe grains appeared less defective than Zn-Mg-RE grains and that the microstructure of these latter grains looks like that of crystals grown by the same technique. Annealing and mechanical polishing effects have also been analysed in the case of Al-Pd-Mn grains. It appeared that annealing improves the quasicrystalline lattice perfection by lowering phason strains insofar as no precipitates are nucleated. Mechanical polishing can introduce defects, located at the external surfaces, having the shape of bands.

Quasiperiodicity in decagonal phases forced by inclined net planes?

It is generally assumed that decagonal quasicrystals show periodically arranged atomic layers only on net planes perpendicular to the tenfold axis and quasiperiodically arranged ones parallel to it. However, there also do exist only slightly puckered atomic layers that are periodically arranged and inclined to the tenfold axis. They coincide with the net planes of the periodic average structures of the decagonal phase and are related to the strongest Bragg reflections. Since they link quasiperiodic and periodic directions, inclined net planes may play a crucial role for growth and stabilization of decagonal quasicrystals. In fact, it is shown how ideal quasiperiodic long-range order and inflation symmetry allow for the existence of inclined net planes with small corrugation and reinforce the relation with the periodic average structures.

Mechanical spectroscopy of quasicrystals

Quasicrystals are solids with long-range quasiperiodic order and orientational order with disallowed crystallographic symmetry (e.g. 5-fold). In addition to the point defects known from crystals (vacancies, interstitials, or antistructure atoms) quasicrystals contain phasonic defects (phasons), which represent violations of the matching rules. Phasons allow displacement of atoms without contribution of vacancies. In icosahedral AlPdMn single quasicrystals two internal friction peaks are observed. Peak A at 370 K ( f=3 Hz) has an activation enthalpy of 0.98 eV and a frequency factor close to the Debye frequency (2×10 15 s −1) and is therefore attributed to point defects, i.e. atomic jumps enabled by phason flips or vacancies. Peak B at 870 K ( f=3 Hz) is determined by an activation enthalpy of 4 eV and an abnormal high frequency factor (3×10 24 s −1), is much broader than a simple Debye-peak, and highly asymmetric. The temperature variation of the relaxation time exhibits significant deviation from ideal Arrhenius behaviour for higher temperatures ( T>920 K) indicating a sort of phase transition. Peak B can be attributed to a relaxation process in which dislocation movement is controlled by creation and movement of phason defects.

Long-range effects on optical absorption in quasiperiodic lattices

We consider exciton optical absorption in quasiperiodic lattices, focusing our attention on the Fibonacci case as a typical example. The absorption spectrum is evaluated by solving numerically the equation of motion of the Frenkel-exciton problem on the lattice, in which on-site energies take on two values according to the Fibonacci sequence. We find that the quasiperiodic order causes the occurrence of well-defined characteristic features in the absorption spectra. We also develop an analytical method that relates satellite lines with the Fourier pattern of the lattice. Our predictions can be used to determine experimentally the long-range quasiperiodic order from optical measurements.

Phonon localization in one-dimensional quasiperiodic chains.

Quasiperiodic long-range order is intermediate between spatial periodicity and disorder, and the excitations in one-dimensional (1D) quasiperiodic systems are believed to be transitional between extended and localized. These ideas are tested with a numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova (FK) and Lennard-Jones (LJ). The ground-state configurations and the eigenfrequencies and eigenfunctions for harmonic excitations are determined. Aubry's transition by breaking the analyticity is observed in the ground state of each model, but the behavior of the excitations is qualitatively different. Phonon localization is observed for some modes in the LJ chain on both sides of the transition. The localization phenomenon apparently is decoupled from the distribution of eigenfrequencies since the spectrum changes from continuous to Cantor-set-like when the interaction parameters are varied to cross the analyticity-breaking transition. The eigenfunctions of the FK chain satisfy the "quasi-Bloch" theorem below the transition, but not above it, while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.

Decagonal quasicrystalline or microcrystalline structures: The specific case of Al-Cu-Co(-Si).

An extensive structural analysis of a decagonal ${\mathrm{Al}}_{63}$${\mathrm{Cu}}_{17.5}$${\mathrm{Co}}_{17.5}$${\mathrm{Si}}_{2}$ alloy is reported. The structure is analyzed within the context of both microcrystalline and quasicrystalline models through the examination of single-crystal x-ray diffraction patterns. We conclude that our sample is in a microcrystalline state formed of coherent twins of an approximant state of the quasicrystal. We underline that diffraction patterns exhibiting sharp peaks and perfect tenfold symmetry do not necessarily correspond to long-range quasiperiodic order, which could have implications for the physical properties of quasicrystal-like alloys. The present work also gives us the opportunity to point out and discuss some general features: (i) the intensity cutoff problem for quasicrystal structure determinations, (ii) the consequence in structure determination if a sample supposed to be a quasicrystal is in fact a microcrystal, and (iii) the proper determination of basis vectors in decagonal quasicrystals.