Comparing light-induced colloidal quasicrystals with different rotational symmetries

Quasicrystals are aperiodic structures with long-range orientational order. Unlike crystals, quasicrystals can, in principle, possess any non-crystallographic rotational symmetry. However, only a few of these rotational symmetries have been observed. By using Monte Carlo simulations of colloidal particles in laser interference patterns with quasicrystalline symmetry, we compare the onset of quasicrystalline order for different rotational symmetries in two dimensions. We find that quasicrystals with 5-, 8-, 10-, and 12-fold rotational axes can be induced with lower laser intensities than quasicrystals with other non-crystallographic rotational symmetries. We relate this finding to the number of local symmetry centers in the respective interference patterns.     

Self-similarity and self-inversion of quasicrystals

The discovery of quasicrystals played a revolutionary role in the condensed matter science and forced to renounce the dogma of the classical crystallography that the regular filling of the space by identical blocks is reduced solely to the Fedorov space groups. It is shown that aperiodic crystals, apart from the similarity, exhibit the self-inversion property. In a broadened sense, the self-inversion implies the possible composition of the inversion with translations, rotations, and homothety, whereas pure reflection by itself in a circle can be absent as an independent symmetry element. It is demonstrated that the symmetry of aperiodic tilings is described by Schottky groups (which belong to a particular type of Kleinian groups generated by the linear fractional Möbius transformations); in the theory of aperiodic crystals, the Schottky groups play the same role that the Fedorov groups play in the theory of crystal lattices. The local matching rules for the Penrose fractal tiling are derived, the problem of choice of the fundamental region of the group of motions of a quasicrystal is discussed, and the relation between the symmetry of aperiodic tilings and the symmetry of constructive fractals is analyzed.     

Aperiodic crystals: A contradictio in terminis?

Although in the prevailing view a necessary condition for having a crystalline phase is lattice periodicity, it has become clear in the last decades that there are physical systems with many properties of the usual crystalline state but without three-dimensional lattice periodicity. Incommensurate modulated crystals have been known now for some time, and a couple of years ago much excitement was raised by the discovery of quasicrystals, systems with long-range order but with five-fold symmetry axes, which exclude lattice periodicity.A discussion is given of the various generalizations of the concept of lattice periodicity. In fact, these go from ordinary periodic crystal st structures to almost chaotic ones. One of these is the notion of quasiperiodicity. Section two deals with a special type of these quasiperiodic systems, tilings or space fillings with tiles or blocks of a small number of types. In section three the symmetry of quasiperiodic systems is discussed. Here the embedding into a higher-dimensional space is the key concept. Section four deals with N-dimensional crystallographic groups that occur as symmetry groups of quasiperiodic systems, so called superspace groups. In section five the diffraction from quasiperiodic systems is treated, and in section six it is shown that in some cases quasiperiodic structures may be approximated by periodic ones, and that periodic systems sometimes are more conveniently described by quasiperiodic ones. The emphasis in the symmetry discussion is on quasicrystals.This is even more so in the remaining sections. Section seven gives a brief account of the many experimental data, section eight describes what is known about the microscopic structure. Imperfections are even more important for quasiperiodic systems than for periodic ones. They are discussed in section nine.Not only microscopically do quasiperiodic systems have similarities with ordinary crystals, but also macroscopically. The morphological laws may be generalized to quasiperiodic systems, as shown in section ten. The consequences of quasiperiodicity on the physical properties is still to a large extent unclear. Mathematically they differ much from periodic systems. A discussion of a number of results is given in section eleven.     

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.     

Symmetry of quasicrystals

The definition of an aperiodic crystal (quasicrystal) as a solid that is characterized by the forbidden symmetry suggests the existence of an unsolved problem, because, in a mutually exclusive manner, it appeals to the fundamental theorem of classical crystallography. Using the Penrose tiling as an example, we have investigated the symmetry properties of aperiodic tilings for the purpose to establish the allowed symmetry groups of quasicrystals. The filling of the Euclidean space according to an aperiodic law is considered as the action of an infinite number of group elements on a fundamental domain in the non-Euclidean space. It is concluded that all locally equivalent tilings have a common “parent” structure and, consequently, the same symmetry group. An idealized object, namely, an infinitely refined tiling, is introduced. It is shown that the symmetry operations of this object are operations of the similarity (rotational homothety). A positive answer is given to the question about a possible composition of operations of the similarity with different singular points. It is demonstrated that the transformations of orientation-preserving aperiodic crystals are isomorphic to a discrete subgroup of the Möbius group PSL(2, ?); i.e., they can be realized as discrete subgroups of the full group of motions in the Lobachevsky space. The problem of classification of the allowed types of aperiodic tilings is reduced to the procedure of enumeration of the aforementioned discrete subgroups.     

一维六方准晶非周期平面内的平面应变理论

针对一维六方准晶中一种新平面弹性问题, 即非周期平面内的平面弹性问题, 通过引入应力势函数, 建立了一维六方准晶中非周期平面内的平面应变理论. 作为应用, 求解了一维六方准晶中垂直于准周期方向的椭圆孔口问题, 得到了其弹性应力场的解析解. 在极限情形下, 可给出裂纹问题的解. 关键词： 一维六方准晶 椭圆孔口 广义解析函数     

Global energy spectra of bands and density of states in a class of two-tile systems

Electronic properties of a general class of one-dimensional two-tile systems are calculated exactly. The systems containing periodic crystals, generalized Fibonacci quasicrystals, generalized Thue-Morse aperiodic lattices and even other two-tile aperiodic lattices, can be divided into two different families which are constructed by the inflation rules: {A, B}{A ^m11 B ^m12,A ^m21 B ^m22} and {A, B}{A ⁿ¹¹ B ⁿ¹²,B ⁿ²¹ A ⁿ²²}, respectively. As typical examples, global spectra of bands and density of states in some two-tile aperiodic systems are numerically calculated. Some interesting properties are obtained.     

Decagonal and Dodecagonal Quasicrystals Obtained by Molecular Dynamics Simulations

Double-well potentials are used for molecular dynamics simulation in monatomic systems. The potentials change as their parameters are adjusted, resulting in different structures. Of particular interest, we obtain decagonal and dodecagonal quasicrystals by simulations. We also verify the results and explain the formation of quasicrystals from the perspective of potential energy.     

Domain wall motion in aperiodic crystal systems and magnetoelectrics

Three classes in the family of aperiodic crystal systems are the displacive modulated phases, the incommensurate composites and quasicrystals. In each of these families one can have a phase transition incommensurate-incommensurate within the incommensurate phase. This transition can be described as a transition from a smooth dependence on the parameters to a discontinuous one. This transition usually is accompanied by the occurrence of a soft excitation that can be considered as a soft phason. The character of the phason and the phase transition is discussed for an example from each class. Finally the situation in a more complicated system with two different subsystems is considered: the case of coupling of a ferroelectric and antiferromagnetic phase.     

How do quasicrystals grow?

Using molecular simulations, we show that the aperiodic growth of quasicrystals is controlled by the ability of the growing quasicrystal nucleus to incorporate kinetically trapped atoms into the solid phase with minimal rearrangement. In the system under investigation, which forms a dodecagonal quasicrystal, we show that this process occurs through the assimilation of stable icosahedral clusters by the growing quasicrystal. Our results demonstrate how local atomic interactions give rise to the long-range aperiodicity of quasicrystals.     

Theoretical considerations about solid-state diffusion of impurities into crystals

Solid-state diffusion of impurities into crystals and quasicrystals is essential for many physical processes concerned with the growth of novel semiconductor materials and the fabrication of electronic/energy devices with commercial viability. Here relevant considerations about diffusion in these systems are presented and discussed.     

Incommensurability in crystals

The aim of this review is to present aperiodic crystals from a unifying point of view, showing why it is justified to call them crystals, despite the lack of a three-dimensional lattice periodicity, and to discuss in what sense they differ from periodic crystals in structure, symmetry and other physical properties. The extension of the concept of crystal has been based, during the last two decades, on investigation of incommensurate crystal phases. Among these, the most important ones are the modulated crystals. Their crystallographic nature, already apparent in the diffraction pattern, could be made explicit on the symmetry level by embedding in a higher-dimensional Euclidean space. The recent discovery of quasicrystal phases (representing a fairly different class of aperiodic crystals than the modulated ones) can also be approached in a similar way. Furthermore, it now appears that another class of incommensurate crystals, the so-called composite structures, represents a kind of intermediate case between the other two.

In § 1 basic concepts are presented, together with a number of compounds given as illustration for typical incommensurate crystal phases. In § 2 we deal with the general formalism allowing a crystallographic symmetry characterization. It is intended as a first approach to crystal-structure determination, which justifies the emphasis on the diffraction pattern and on the modulated-crystal case. The appropriate generalization to the quasicrystal case is considered in § 3. The crystallographic nature of the incommensurate phases is apparent in their growth forms. It has been known for centuries that crystal morphology is essentially based on lattice periodicity. It is therefore fascinating to discover how nature solves the problem in the incommensurate crystal case; we discuss this in § 4. The origin of incommensurability is the subject of § 5 (on a phenomenological level) and of § 6 (on a microscopic level). In § 7 the basic concepts needed in crystal-structure determination are discussed in more detail than was appropriate in § 2. In § 8 we discuss those physical properties which are more closely related to incommensurability, fitting nevertheless into the general frame of crystal physics. In § 9 we deal with defects, again in particular with the additional ones due to the more complex structure of aperiodic crystals, making clear at the same time why defects play an even more important role than in periodic crystals.     

Hexagonally poled lithium niobate: A two-dimensional nonlinear photonic crystal

We report on the fabrication of what we believe is the first example of a two-dimensional (2D) nonlinear photonic crystal [Berger, Phys. Rev. Lett. 81, 4136 (1998)], where the refractive index is constant but where the 2nd order nonlinear susceptibility is spatially periodic. Such crystals allow for efficient quasi-phase-matched 2nd harmonic generation using multiple reciprocal lattice vectors. External 2nd harmonic conversion efficiencies >60% were measured with picosecond pulses. The fabrication technique is extremely versatile and should allow for the fabrication of a broad range of 2D crystals including quasicrystals.     

A quantum field theory of lattice dynamics and melting in nonprimitive crystals

A unified theory of lattice dynamics and melting is presented from the view-point of quantum field theory at finite temperature. The lattice dynamics and the melting in nonprimitive crystals which contain more than one atom in a unit cell are investigated in the random phase approximation using the two-bands model for atoms. The gap equation for atoms plays an essential role in constructing the lattice dynamics and determining the melting temperature. Our theory is applied to sphalerite and diamond structure in the nearest-neighbour approximation for the interatomic potentials. Our theory covers all types of crystals.     

A new method for generation of quasi-periodic structures withn fold axes: Application to five and seven folds

A new geometrical method for generating aperiodic lattices forn-fold non-crystallographic axes is described. The method is based on the self-similarity principle. It makes use of the principles of gnomons to divide the basic triangle of a regular polygon of 2n sides to appropriate isosceles triangles and to generate a minimum set of rhombi required to fill that polygon. The method is applicable to anyn-fold noncrystallographic axis. It is first shown how these regular polygons can be obtained and how these can be used to generate aperiodic structures. In particular, the application of this method to the cases of five-fold and seven-fold axes is discussed. The present method indicates that the recursion rule used by others earlier is a restricted one and that several aperiodic lattices with five fold symmetry could be generated. It is also shown how a limited array of approximately square cells with large dimensions could be detected in a quasi lattice and these are compared with the unit cell dimensions of MnAl₆ suggested by Pauling. In addition, the recursion rule for sub-dividing the three basic rhombi of seven-fold structure was obtained and the aperiodic lattice thus generated is also shown. Based on the lecture by the author “Quasi crystals: Is Linus Pauling right” and delivered on 16 December 1985 and arranged by the Departments of Physics, Metallurgy, Materials Research Laboratory, and Instrumentation Services Unit, Indian Institute of Science, Bangalore.     

Self-assembly of monatomic complex crystals and quasicrystals with a double-well interaction potential

For the study of crystal formation and dynamics, we introduce a simple two-dimensional monatomic model system with a parametrized interaction potential. We find in molecular dynamics simulations that a surprising variety of crystals, a decagonal, and a dodecagonal quasicrystal are self-assembled. In the case of the quasicrystals, the particles reorder by phason flips at elevated temperatures. During annealing, the entropically stabilized decagonal quasicrystal undergoes a reversible phase transition at 65% of the melting temperature into an approximant, which is monitored by the rotation of the de Bruijn surface in hyperspace.     

Superconducting vortex pinning with artificial magnetic nanostructures

This review is dedicated to summarizing the recent research on vortex dynamics and pinning effects in superconducting films with artificial magnetic structures. The fabrication of hybrid superconducting/magnetic systems is presented together with the wide variety of properties that arise from the interaction between the superconducting vortex lattice and the artificial magnetic nanostructures. Specifically, we review the role that the most important parameters in the vortex dynamics of films with regular array of dots play. In particular, we discuss the phenomena that appear when the symmetry of a regular dot array is distorted from regularity towards complete disorder including rectangular, asymmetric, and aperiodic arrays. The interesting phenomena that appear include vortex-lattice reconfigurations, anisotropic dynamics, channeling, and guided motion as well as ratchet effects. The different regimes are summarized in a phase diagram indicating the transitions that take place as the characteristic distances of the array are modified respect to the superconducting coherence length. Future directions are sketched out indicating the vast open area of research in this field.     

Molecular dynamics simulations of laser induced surface melting in orthorhombic Al13Co4

Laser induced surface melting of the aluminum–cobalt alloy Al₁₃Co₄ is investigated. For the simulations of the lattice ions we use molecular dynamics, while for the time evolution of the electron temperature a generalized heat-conduction equation is solved. Energy transfer between the sub-systems is allowed by an electron–phonon coupling term. This combined treatment of the electronic and atomic systems is an extension of the well-known two-temperature model [Anisimov et al. in JETP Lett. 39(2), 1974]. The alloy shows large structural affinity to decagonal quasicrystals, which have an in-plane five-fold symmetry,while in perpendicular direction the planes are stacked periodically. As a consequence we observe slight anisotropic melting behavior.     

Achieving epitaxy between incommensurate materials by quasicrystalline interlayers

Epitaxial interfaces of commensurate periodic materials can be characterized by a locking into registry of their atomic structure. This characteristic is identified as a natural framework to capture the essence of epitaxy also for systems including quasicrystalline materials. The resulting general definition for epitaxy requires a matching of reciprocal lattice points. The consequences for the real space structure of an epitaxial interface between quasiperiodic and periodic materials are explored and an experimental realization of such an interface is presented. It is demonstrated that due to their higher number of reciprocal lattice basis vectors (exceeding three), quasicrystals can provide interlayers epitaxially linking incommensurate materials.