Two-dimensional Quasicrystals Research Articles

Three-dimensional Green's functions for two-dimensional quasi-crystal bimaterials

Owing to their specific structure, which can neither be classified as crystalline nor amorphous, quasi-crystals (QCs) exhibit properties that are interesting to both material science and mathematical physics or continuum mechanics. Within the framework of a mathematical theory of elasticity, one major focus is on features evolving from the coupling of phonon and phason fields, which is not observed in classical crystalline or amorphous materials. This paper deals with the problems of combinations of point phonon forces and point phason forces, which are applied to the interior of infinite solids and bimaterial solids of two-dimensional hexagonal QCs. By using the general solution of QCs, a series of displacement functions is adopted to obtain the analytical results when the two half-spaces are supposed to be ideally bonded or to be in smooth contact. In the final expressions, we provide three-dimensional Green’s functions for infinite bimaterial QC solids in the closed form, which are very convenient to be used in the study of dislocations, cracks and inhomogeneities of the new solid phase. Furthermore, the paper is concluded by a discussion of some special cases, in which Green’s functions for infinite transversely isotropic solids and Green’s functions for a half-space with free or fixed boundary are given.

A general solution of elasto-hydrodynamics of two-dimensional quasicrystals

Dynamic problems of quasicrystals are analysed within the framework of continuum mechanics. Phonon excitations yield wave propagation while phason excitations yield atomic diffusion. Phonon fields obey the equations of motion and phason fields obey the diffusion equations. Governing equations of elasto-hydrodynamics of decagonal quasicrystals combine characteristics of the equations of motion and diffusion. A general solution is derived in terms of two introduced auxiliary functions. It provides an easy-to-use approach to solve initial-boundary value problems encountered in elasticity problems for quasicrystals. Explicit expressions for displacements and stresses in the phonon and phason fields can be directly obtained. A derived general solution is also applicable to some planar quasicrystals such as pentagonal quasicrystals with fivefold symmetry.

Dynamics of particle flips in two-dimensional quasicrystals

The dynamics of quasicrystals is more complicated than the dynamics of periodic solids and difficult to study in experiments. Here, we investigate a decagonal and a dodecagonal quasicrystal using molecular dynamics simulations of the Lennard-Jones-Gauss interaction system. We observe that the short time dynamics is dominated by stochastic particle motion, so-called phason flips, which can be either single-particle jumps or correlated ring-like multi-particle moves. Over long times, the flip mechanism is efficient in reordering the quasicrystals and can generate diffusion. The temperature dependence of diffusion is described by an Arrhenius law. We also study the spatial distribution and correlation of mobile particles by analyzing the dynamic propensity.

Drag Reduction Effect of Dimples Arranged in Two-Dimensional Quasicrystal Structure

Rotating fluids experiments were carried out by CAP2000+ cone viscometer, to examine the drag reduction properties of dimples arranged in quasicrystal structure. The dimples were fabricated on the surface of duralunmin (LY12) plates. Compared with the periodic arrays, the dimples arranged in quasicrystal structure, especially the 12-fold quasicrystal structure, could significantly reduce the wall shear stress. And the relative drag reduction efficiency changes periodically with the depth of dimple. Flow-visualization experiment verified that the coherent effect of dimples arranged in quasicrystal structure and the fluids could efficiently inhibit the extending intensity of radial secondary flow, which strengthens the drag reduction effect.

Čerenkov-type second-harmonic generation with fundamental beams of different polarizations

We analyze the conical second-harmonic generation via Cerenkov-type phase matching in two-dimensional decagonal nonlinear photonic quasi-crystals for linear, circular, and elliptical polarized fundamental beams. Depending on the polarization state, we observe either centrosymmetric or noncentrosymmetric intensity modulation of the second-harmonic rings. We explain this effect by employing the concept of effective quadratic nonlinearity combined with the hexagonal symmetry of the individual ferroelectric domains.

Two-photon fabrication of photonic crystals by single-beam laser holographic lithography

We demonstrate a novel approach for the easy fabrication of photonic crystals (PCs) with arbitrary optical lattice using two-photon holographic lithography (HL). The quadratic dependence of the two-photon absorption process dramatically increases the contrast of the pattern and the low absorption coefficient of the photoresist at 720 nm lead to a high aspect ratio and low fluctuation of the pattern. By applying such a technique, defect-free and highly uniform one-dimensional gratings, two-dimensional quasicrystals, and three-dimensional (3D) face center cube-type PCs of high aspect ratio and small filling factor are obtained. This two-photon based HL technique is particularly suitable for making 3D PCs and quasicrystals with complex and fine structural details.

Resonant transmission of electromagnetic waves through two-dimensional photonic quasicrystals

We present numerical simulations of electromagnetic millimeter-wave propagation in a two-dimensional lattice of dielectric rods arranged in a tenfold Penrose tiling. We find (i) isotropic photonic band gap as expected for quasicrystals and (ii) localized states. We demonstrate that the high frequency edge of the second band gap is characterized by a very small refractive index (fast light). We study the transmission of electromagnetic waves in the frequency range corresponding to fast light and demonstrate that it is related to tunneling through localized states. We use the fast light phenomenon to design a focusing device—a planoconcave lens.

Localization characteristics of two-dimensional quasicrystals consisting of metal nanoparticles

Using the eigen-decomposition method, we investigated the plasmonic modes in a two-dimensional quasicrystalline array of metal nanoparticles. Various properties of the plasmonic modes, such as their symmetry, radiation loss and spatial localization are studied. Some collective plasmonic modes are found to be leaky with out-of-plane radiation loss, but some modes can have very high fidelity. To facilitate the analysis on the complex behavior of the collective plasmonic modes, we considered a trajectory map in a three dimension parameter space defined by the frequency of the mode, the participation ratio of the mode that gives the spatial distribution, and the eigenvalue that is related to the inverse lifetime of the mode. With the help of the trajectory map method, we found that there are modes that are localized in some special ways and the plasmonic mode with highest spatial localization and highly fidelity is found to be a type of anti-phase ring mode. In general, different localized modes have very different radial decay behaviors. There is no special relationship between the fidelity of the modes and their spatial localization.

Complete band gaps in two-dimensional photonic quasicrystals

We introduce a novel optimization method to design the first examples of photonic quasicrystals with substantial, complete photonic band gaps (PBGs): that is, a range of frequencies over which electromagnetic wave propagation is forbidden for all directions and polarizations. The method can be applied to photonic quasicrystals with arbitrary rotational symmetry; here, we illustrate the results for 5- and 8-fold symmetric quasicrystals. The optimized band gaps are highly isotropic, which may offer advantages over photonic crystals for certain applications.

Interaction of fast electron beam with photonic quasicrystals

The interactions of two-dimensional 10-fold and 12-fold photonic quasicrystals with a moving electron beam have been studied by using the multiple-scattering method. The electron energy loss spectroscopy (EELS) and three-dimensional local density of states in these systems have been calculated. Some three-dimensional localized states in these defect-free two-dimensional systems have been found. It has been demonstrated that these localized states can be explored by means of the EELS.

STUDY ON ELASTIC ANALYSIS OF CRACK PROBLEM OF TWO-DIMENSIONAL DECAGONAL QUASICRYSTALS OF POINT GROUP 10, $\overline {10}$

By introducing a stress potential function, we transform the plane elasticity equations of two-dimensional quasicrystals of point group 10, [Formula: see text] to a partial differential equation. And then we use the complex variable function method for classical elasticity theory to that of the quasicrystals. As an example, a decagonal quasicrystal in which there is an arc is subjected to a uniform pressure p in the elliptic notch of the decagonal quasicrystal is considered. With the help of conformal mapping, we obtain the exact solution for the elliptic notch problem of quasicrystals. The work indicates that the stress potential and complex variable function methods are very useful for solving the complicated boundary value problems of higher order partial differential equations which originate from quasicrystal elasticity.

Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography

Two-dimensional (2D) photonic quasicrystal (PQC) template patterns have been fabricated by using multiple-exposure holographic lithography (MHL). The MHL technique is based on Lloyd’s mirror setup using interference of two equivalent beams, which is an excellent method for fabricating periodic structures of a large area. The 2D PQC array shows various sizes and shapes with a change in the incident angle (θ) and rotation angle (γ) of the multiple-exposure beams. In addition, the filling factor (η), which is a very important factor for optical characteristic of 2D PQCs, was controlled by the exposure time (τ). The fabricated PQCs showed eight-, ten-, and 12-fold patterns with high-order rotational symmetries. Diffraction patterns using 632.8 nm HeNe laser were also observed with n-rotation symmetry for the corresponding n-fold PQCs. After fabricating the PQCs, reactive-ion etching of the silicon wafer was performed with 50-nm-thick chromium (Cr) as a mask in CF4/O2 plasma to a 1.3 μm depth. The authors confirmed that fabrication of n-fold 2D PQCs in a large area with suitable optical characteristics could be controlled by some parameters of MHL.

Antiferromagnetism in two-dimensional quasicrystals

Abstract Quasiperiodic structures possess long range positional order, but are freed of constraints imposed by translational invariance. For spins interacting via Heisenberg couplings, one may expect therefore to find novel magnetic configurations in such structures. We have studied magnetic properties for simple two dimensional models, as a first step towards understanding experimentally studied magnetic quasicrystals such as the Zn–Mg–R(rare earth) compounds. We analyse properties of the antiferromagnetic groundstate and magnon excitation modes for bipartite tilings such as the octagonal and Penrose tilings. In the absence of frustration, one has an inhomogeneous Neel-ordered ground state, with local quantum fluctuations of the local staggered magnetic order parameter. We study the spin wave spectrum and wavefunctions of such antiferromagnets within the linear spin wave approximation. Some results for spin-spin correlations in these systems are discussed.

General solutions of three-dimensional problems for two-dimensional quasicrystals

Three-dimensional problems are systematically investigated for the coupled equations in two-dimensional hexagonal quasicrystals, and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) solutions and generalized Elliott–Lodge (E–L) solutions, are presented, respectively. By introducing two higher-order displacement functions, an operator analysis technique is applied in a novel way to obtain generalized LHN solutions. For further simplification, a decomposition and superposition procedure is taken to replace the higher-order displacement functions with five quasi-harmonic displacement functions, and then generalized E–L solutions are simplified in terms of these functions. In consideration of different cases of characteristic roots, generalized E–L solutions take different forms, but all are in simple forms that are conveniently applied. To illustrate the application of the general solutions obtained, the closed form solution is obtained for an infinite quasicrystal medium subjected to a point force at an arbitrary point.

Affine symmetry groups in 2D-quasicrystals

This paper is devoted to the mathematical description of possible finite affine symmetry groups in two-dimensional quasicrystals.

A Theory of General Solutions of Plane Problems in Two-Dimensional Octagonal Quasicrystals

A theory of general solutions of plane problems is developed for the coupled equations in plane elasticity of two-dimensional octagonal quasicrystals. In virtue of the operator method, the general solutions of the antiplane and inplane problems are given constructively with two displacement functions. The introduced displacement functions have to satisfy higher order partial differential equations, and therefore it is difficult to obtain rigorous analytic solutions directly and is not applicable in most cases. In this case, a decomposition and superposition procedure is employed to replace the higher order displacement functions with some lower order displacement functions, and accordingly the general solutions are further simplified in terms of these functions. In consideration of different cases of characteristic roots, the general solution of the antiplane problem involves two cases, and the general solution of the inplane problem takes three cases, but all are in simple forms that are convenient to be applied. Furthermore, it is noted that the general solutions obtained here are complete in x3-convex domains.

Engineering two-dimensional nonlinear photonic quasi-crystals

A known algorithm for modeling quasi-periodic lattices is used to generate two-dimensional quadratic nonlinear photonic quasi-crystals containing a set of desired discrete spectral components. This allows us to fabricate optical devices in which an arbitrary set of nonlinear optical processes can be efficient. We demonstrate this capability by fabricating two devices: a multidirectional single-frequency doubler and a multidirectional, multifrequency doubler that is capable of nearly collinear doubling of cw radiation in the optical communication C band (1530-1570 nm) through angle tuning.

Elastic properties of two-dimensional quasicrystals

Quasicrystals (QC) with two-dimensional quasiperiodic and one-dimensional periodic structure are considered. Their symmetry can be described by embedding the three-dimensional physical space V(E) in a five-dimensional superspace V, which is the direct sum of V(E) and a two-dimensional internal space V(I). A displacement v in V can be written as v = u + w, where u in V(E) and w in V(I). If the QC has a point group P in V(E) that is crystallographic, it is assumed that w and a vector u' in V(E) lying in the plane in which the crystal is quasiperiodic transform under equivalent representations of P, inequivalent ones if the point group is 5-, 8-, 10- or 12-gonal. From the Neumann principle follow restrictions on the form of the phonon, phason and phonon-phason coupling contributions to the elastic stiffness matrix that can be determined by combining the restrictions obtained for a set of elements generating the point group of interest. For the phonon part, the restrictions obtained for the generating elements do not depend on the system to which the point group belongs. This remains true for the phason and coupling parts in the case of crystallographic point groups but, in general, breaks down for the non-crystallographic ones. The form of the symmetric 12 x 12 matrix giving the phonon, phason and phonon-phason coupling contributions to the elastic stiffness is presented in graphic notation.

Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals

We study some typical defect problems in one-dimensional (1D) hexagonal and two-dimensional (2D) octagonal quasicrystals. The first part of this investigation addresses in detail a uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. A general solution is derived in terms of two functions φ1, φ2, which satisfy wave equations, and another harmonic function φ3. Elementary expressions for the phonon and phason displacements, strains, stresses, electric potential, electric fields and electric displacements induced by the moving screw dislocation are then arrived at by employing the obtained general solution. The derived solution is verified by comparison with existing solutions. Also obtained in this part of the investigation is the total energy of the moving screw dislocation. The second part of this investigation is devoted to the study of the interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal. Here the crack penetrates through the solid along the period direction and the dislocation line is parallel to the period direction. We first derive a general solution in terms of four analytic functions for plane strain problem in octagonal quasicrystals by means of differential operator theory and the complex variable method. All the phonon and phason displacements and stresses can be expressed in terms of the four analytic functions. Then we derive the exact solution for a straight dislocation near a semi-infinite crack in an octagonal quasicrystal, and also present the phonon and phason stress intensity factors induced by the straight dislocation and remote loads.

Remarks on symmetries of 2D-quasicrystals

The paper presents a strict mathematical proof of the classifications of finite groups of symmetries of two-dimensional quasicrystals given by Hermisson, Richard and Baake in 1997.