Quasiperiodic Crystals Research Articles

Propagation of acoustic waves in 2D periodic and quasiperiodic phononic crystals

In this study, we have investigated theoretically the propagation of longitudinal sonic waves in 2D periodic and aperiodic (quasiperiodic) phononic crystals. Fibonacci sequence has been used to generate the aperiodic structure. The effect of transformation from 2D periodic to aperiodic structure has been elucidated by calculating the transmission coefficient. The consequences of inserting a circular solid cylinder inside a host water matrix have been repeated at the same calculating conditions. Changing filling fraction has been used to tune the stop band width.

Efficient Modulation of Spin Waves in Two-Dimensional Octagonal Magnonic Crystal.

Efficient tunability of magnonic spectra is demonstrated in two-dimensional ferromagnetic antidot lattices with different lattice constants arranged in the octagonal lattice which can be considered as quasi-periodic magnonic crystals due to the presence of broken translational symmetry. The precessional dynamics of these samples are investigated in the frequency domain with the help of broadband ferromagnetic resonance spectrometer. A rich variation in the spin wave spectra is observed with the variation of lattice constant as well as the strength and orientation of the bias magnetic field. A broad band of spin wave modes are observed for the denser array, which finally converges to two spin wave modes for the sparsest one. In addition to this, the most intense spin wave frequency shows an 8-fold anisotropy with a superposition of weak 4- and 2-fold anisotropy, which arises due to the angular variation of the magnetostatic field distribution at different regions of the octagonal lattice. Micromagnetic simulations qualitatively reproduce the experimentally observed modes, and the simulated mode profiles reveal the presence of different types of extended and quantized standing spin wave modes in these samples. The observations are important for the tunable and anisotropic propagation of spin waves in magnonic crystal based devices.

Crystallography and dynamics in superspace

The discovery in materials of long range order without lattice periodicity has generated a very broad field of research with the introduction of totally new concepts. The common feature of this new state of matter is the translational symmetry recovered by adding supplementary dimensions to the physical space, defining a crystallographic superspace. Along these supplementary directions, specific low excitations, so-called phasons, are predicted and observed. These quasi-periodic crystals may present phase transitions as function of external parameters, which have then to be described as symmetry breakings within these crystallographic superspaces. Three different families of quasi-periodic crystals are usually considered: incommensurate modulated crystals, aperiodic composites and quasicrystals. Their crystallography and dynamics are discussed.

Pushing the limits of crystallography.

A very serious concern of scientists dealing with crystal structure refinement, including theoretical research, pertains to the characteristic bias in calculated versus measured diffraction intensities, observed particularly in the weak reflection regime. This bias is here attributed to corrective factors for phonons and, even more distinctly, phasons, and credible proof supporting this assumption is given. The lack of a consistent theory of phasons in quasicrystals significantly contributes to this characteristic bias. It is shown that the most commonly used exponential Debye-Waller factor for phasons fails in the case of quasicrystals, and a novel method of calculating the correction factor within a statistical approach is proposed. The results obtained for model quasiperiodic systems show that phasonic perturbations can be successfully described and refinement fits of high quality are achievable. The standard Debye-Waller factor for phonons works equally well for periodic and quasiperiodic crystals, and it is only in the last steps of a refinement that different correction functions need to be applied to improve the fit quality.

Band structures of elastic waves in two-dimensional eight-fold solid–solid quasi-periodic phononic crystals**Project supported by the National Natural Science Foundation of China (Grant Nos. 11272043 and 10902012) and the Project-sponsored by SRF for ROCS, SEM.

Combined with the supercell method, band structures of the anti-plane and in-plane modes of two-dimensional (2D) eight-fold solid–solid quasi-periodic phononic crystals (QPNCs) are calculated by using the finite element method. The influences of the supercell on the band structure and the wave localization phenomenon are discussed based on the modal distributions. The reason for the appearance of unphysical bands is analyzed. The influence of the incidence angle on the transmission spectrum is also discussed.

Application of the multiband -models to quantum transport in quasi-periodic crystals

We consider the quantum transport of a particle in a crystal where the global three-dimensional translation symmetry is removed. Our analysis concerns solid state structures where the microscopic potential of the ions is no longer periodic. This includes materials with variable chemical composition, intergrowth compounds and abrupt junctions. The application of the kp multiband envelope function models to the description of the quantum mechanical motion in such a quasi-periodic structure is investigated. By using a spectral decomposition technique, we show that when the interatomic distance is asymptotically small, the particle probability density can be obtained by the envelope kp model with variable coefficients.

Quasi-periodic crystals – a paradigm shift in crystallography

Crystallography has been one of the mature sciences. Over the years, the modern science of crystallography that started by experimenting with x-ray diffraction from crystals in 1912, has developed a major paradigm – that all crystals are ordered and periodic. Indeed, this was the basis for the definition of "crystal" in textbooks of crystallography and x-ray diffraction. Based upon a vast number of experimental data, constantly improving research tools, and deepening theoretical understanding of the structure of crystalline materials no revolution was anticipated in our understanding the atomic order of solids. However, such revolution did happen with the discovery of the Icosahedral phase, the first quasi-periodic crystal (QC) in 1982, and its announcement in 1984 [1, 2]. QCs are ordered materials, but their atomic order is quasiperiodic rather than periodic, enabling formation of crystal symmetries, such as icosahedral symmetry, which cannot exist in periodic materials. The discovery created deep cracks in this paradigm, but the acceptance by the crystallographers' community of the new class of ordered crystals did not happen in one day. In fact it took almost a decade for QC order to be accepted by most crystallographers. The official stamp of approval came in a form of a new definition of "Crystal" by the International Union of Crystallographers. The paradigm that all crystals are periodic has thus been changed. It is clear now that although most crystals are ordered and periodic, a good number of them are ordered and quasi-periodic. While believers and nonbelievers were debating, a large volume of experimental and theoretical studies was published, a result of a relentless effort of many groups around the world. Quasi-periodic materials have developed into an exciting interdisciplinary science. This talk will outline the discovery of QCs and describe the important role of electron microscopy as an enabling discovery tool.

Homogenization limit for a multiband effective mass model in heterostructures

We study the homogenization limit of a multiband model that describes the quantum mechanical motion of an electron in a quasi-periodic crystal. In this approach, the distance among the atoms that constitute the material (lattice parameter) is considered a small quantity. Our model include the description of materials with variable chemical composition, intergrowth compounds, and heterostructures. We derive the effective multiband evolution system in the framework of the kp approach. We study the well posedness of the mathematical problem. We compare the effective mass model with the standard kp models for uniform and non-uniforms crystals. We show that in the limit of vanishing lattice parameter, the particle density obtained by the effective mass model, converges to the exact probability density of the particle.

Study on band gap properties of two-dimensional 8-fold quasi-periodic phononic crystals

For the two-dimensional (2D) 8-fold solid-fluid quasi-periodic photonic nanocrystal (QPNC) (steel-water system), the transmission spectra of the systems with different sizes are obtained by experimentation. At the same time, combined with the supercell method, the finite element method (FEM) is used to calculate the dispersion curves and transmission spectrum of the system. The transmission spectra obtained by means of these two methods are in good agreement with each other, but the band gaps appearing in the transmission spectrum calculated using the FEM are clearer than those obtained via experimentation. The dispersion curves of the 2D 8-fold solid-fluid QPNC can be well studied by using the supercell which only contains its center puzzle.

Quasi-Periodic Crystals—The Long Road from Discovery to Acceptance

Three surprising discoveries on the nature of matter and its properties were published in the mid-1980s. All these discoveries led to Nobel prizes. The first discovery was in 1985, when fullerenes were discovered by Robert F. Curl Jr, Sir Harold W. Kroto, and Richard E. Smalley. Fullerenes, also known as buckyballs, are spherical molecules composed of carbon atoms. The discovery of fullerenes launched the field of nano-materials, one of the fastest-growing fields in chemistry today. In 1996, 11 years after the publication of the discovery, the three researchers were jointly awarded the Nobel Prize in chemistry. No controversy surrounded this discovery. In 1986, two IBM researchers, Karl Muller and Johannes Bednorz, discovered high-temperature superconductive materials. Although superconductivity was first discovered in 1911, nobody expected to see this phenomenon at the relatively high temperatures of liquid nitrogen. In 1987, one year after publishing their discovery, the two researchers were awarded the Nobel Prize in physics. Again, no controversy surrounded this discovery, and, as the short period of time between the discovery and awarding of the prize shows, the discovery was enthusiastically embraced by the scientific community. Publication of the third discovery pre-dates the publication of the other two discoveries. I published the discovery of quasi-periodic crystals in 1984 and was awarded a Nobel Prize in 2011, 27 years after the discovery. Unlike the previous two discoveries, this discovery was met with fierce opposition and a substantial amount of controversy. What was so controversial about this discovery that it raised the antagonism of so many people in the scientific community? Why would Linus Pauling, a twice-awarded Nobel Laureate and one of the greatest chemists of the twentieth century, state: “There is no such thing as quasi-crystals, only quasi-scientists”? In order to answer these questions, I must first give a short introduction to crystallography. For that purpose, I will define three basic terms in crystallography: order, periodicity, and rotational symmetry.

Ultrafast filling of an electronic pseudogap in an incommensurate crystal

We investigate the quasiperiodic crystal (LaS)1.196(VS2) by angle and time resolved photoemission spectroscopy. The dispersion of electronic states is in qualitative agreement with band structure calculated for the VS2 slab without the incommensurate distortion. Nonetheless, the spectra display a temperature dependent pseudogap instead of quasiparticles crossing. The sudden photoexcitation at 50 K induces a partial filling of the electronic pseudogap within less than 80 fs. The electronic energy flows into the lattice modes on a comparable timescale. We attribute this surprisingly short timescale to a very strong electron-phonon coupling to the incommensurate distortion. This result sheds light on the electronic localization arising in aperiodic structures and quasicrystals.

Dispersion Engineering for Multifunctional Photonic Crystal Based Nanophotonic Devices at Infrared Wavelengths

TIn this paper, we report the design, the fabrication and the near field optical microscopy of Negative Index Material (NIM) and GRadient INdex (GRIN) photonic crystal based flat lenses. They were fabricated on the basis of an InPbased photonic crystal technological platform including hole and pillar networks fabrication at nanometer scale. They show the possibility of sub-wavelength focusing by all dielectric periodic or quasi-periodic crystals. Particular attention is paid to the analysis of SNOM images using three-dimensional simulations. Finally, in order to demonstrate the versatility of our approach, a two-dimensional cloaking device mixing hole and pillar arrays is evaluated to pave the way for future integrated nanophotonic devices with complex functionalities.

2D quasiperiodic plasmonic crystals

Nanophotonic structures with irregular symmetry, such as quasiperiodic plasmonic crystals, have gained an increasing amount of attention, in particular as potential candidates to enhance the absorption of solar cells in an angular insensitive fashion. To examine the photonic bandstructure of such systems that determines their optical properties, it is necessary to measure and model normal and oblique light interaction with plasmonic crystals. We determine the different propagation vectors and consider the interaction of all possible waveguide modes and particle plasmons in a 2D metallic photonic quasicrystal, in conjunction with the dispersion relations of a slab waveguide. Using a Fano model, we calculate the optical properties for normal and inclined light incidence. Comparing measurements of a quasiperiodic lattice to the modelled spectra for angle of incidence variation in both azimuthal and polar direction of the sample gives excellent agreement and confirms the predictive power of our model.

Defect-free localized modes and coupled-resonator acoustic waveguides constructed in two-dimensional phononic quasicrystals

The localization of acoustic waves in defect-free two-dimensional quasiperiodic phononic crystals has been investigated by using an exact multiple-scattering method. Different types of localized states have been found in eightfold, tenfold, and twelvefold two-dimensional phononic quasicrystals without introducing any defect. The defect-free coupled-resonator acoustic waveguides, based on these localized modes, have been designed, and the resonant transmission properties of acoustic waves in the waveguides have been demonstrated by numerical simulations. It is anticipated that our findings can be potentially applied to phononic devices.

Fractal spectra in generalized Fibonacci one-dimensional magnonic quasicrystals

In this work we carry out a theoretical analysis of the spectra of magnons in quasiperiodic magnonic crystals arranged in accordance with generalized Fibonacci sequences in the exchange regime, by using a model based on a transfer-matrix method together random-phase approximation (RPA). The generalized Fibonacci sequences are characterized by an irrational parameter σ(p,q), which rules the physical properties of the system. We discussed the magnonic fractal spectra for first three generalizations, i.e., silver, bronze and nickel mean. By varying the generation number, we have found that the fragmentation process of allowed bands makes possible the emergence of new allowed magnonic bulk bands in spectra regions that were magnonic band gaps before, such as which occurs in doped semiconductor devices. This interesting property arises in one-dimensional magnonic quasicrystals fabricated in accordance to quasiperiodic sequences, without the need to introduce some deferent atomic layer or defect in the system. We also make a qualitative and quantitative investigations on these magnonic spectra by analyzing the distribution and magnitude of allowed bulk bands in function of the generalized Fibonacci number Fn and as well as how they scale as a function of the number of generations of the sequences, respectively.

Wave propagation in one-dimensional solid–fluid quasi-periodic and aperiodic phononic crystals

The propagation of the elastic waves in one-dimensional (1D) solid–fluid quasi-periodic phononic crystals is studied by employing the concept of the localization factor, which is calculated by the transfer matrix method. The solid–fluid interaction effect at the interfaces between the solid and the fluid components is considered. For comparison, the periodic systems and aperiodic Thue–Morse sequence are also analyzed in this paper. The splitting phenomenon of the pass bands and bandgaps are discussed for these 1D solid–fluid systems. At last the influences of the material impedance ratios on the band structures of the 1D solid–fluid quasi-periodic phononic crystals arranged as Fibonacci sequence are discussed.

Partial band gaps in magnonic crystals

In this work we investigate magnonic band gaps, in the THz frequency range, in periodic and quasiperiodic (Fibonacci sequence) magnonic crystals formed by layers of cobalt and permalloy. Our theoretical model is based on a magnetic Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation. For periodic arrangements, the bulk band structure is analogous to those found in photonic crystals, while for quasiperiodic multilayers it presents additional pass bands similar to those found in photonic crystals with defects.

Band gaps in the terahertz frequency range in quasiperiodic one-dimensional magnonic crystals

In this work we investigate magnonic band gaps, in the terahertz (THz) frequency range, in periodic and quasiperiodic (Fibonacci sequence) magnonic crystals formed by layers of Cobalt (Co) and Permalloy (Py). Our theoretical model is based on a magnetic Heisenberg Hamiltonian in the exchange regime, together with a transfer-matrix treatment within the random-phase approximation (RPA). For periodic arrangements the bulk band structure is analogous to those found in photonic crystals, while for quasiperiodic multilayers it presents additional pass bands similar to those found in doped electronic materials.

Periodic, quasi‐periodic, and random quadratic nonlinear photonic crystals

AbstractQuadratic nonlinear photonic crystals are materials in which the second order susceptibility χ(2) is spatially modulated while the linear susceptibility remains constant. These structures are significantly different than the more common photonic crystals, in which the linear susceptibility is modulated. Nonlinear processes in nonlinear photonic crystals are governed by the phase matching requirements, which are determined by the reciprocal lattice of these crystals. Therefore, the modulation of the nonlinear susceptibility enables to engineer the spatial and spectral response in various three‐wave mixing processes. It enables to support the efficient generation of new optical frequencies at multiple directions. We analyze three wave mixing processes in nonlinear photonic crystals in which the modulation is either periodic, quasi‐periodic, radially symmetric or even random. We discuss both one‐dimensional and two‐dimensional modulations. In addition to harmonic generations, we outline several new possibilities for all‐optical control of the spatial and polarization properties of optical beams in specially designed nonlinear photonic crystals.

Crystal–quasicrystal transition in the Al–Cu–Fe system: Analysis of the local atomic structure

Quasicrystals (quasiperiodic crystals) are attracting intense interest due to their unique properties being non-typical for their crystalline and amorphous phase analogues. Changes of the local structure around Al, Cu, Fe atoms from the quasicrystalline to the crystalline phase transition have been studied. Investigation of the local atomic structure of the Al 65Cu 22Fe 13 quasicrystal and its crystal analogue has been performed on the basis of the (XANES) X-ray absorption near edge structure analysis. Analysis of experimental XANES spectra permits to determine the 3D local atomic structure of the given materials. Theoretical analysis of the experimental data has been carried out on the basis of a self-consistent, real space multiple scattering method (FEFF8.4 code) and a finite difference method (FDMNES2009 code).