Two-dimensional Bravais Lattices Research Articles

Quantum-dot supercrystals for future nanophotonics

The study of supercrystals made of periodically arranged semiconductor quantum dots is essential for the advancement of emerging nanophotonics technologies. By combining the strong spatial confinement of elementary excitations inside quantum dots and exceptional design flexibility, quantum-dot supercrystals provide broad opportunities for engineering desired optical responses and developing superior light manipulation techniques on the nanoscale. Here we suggest tailoring the energy spectrum and wave functions of the supercrystals' collective excitations through the variation of different structural and material parameters. In particular, by calculating the excitonic spectra of quantum dots assembled in two-dimensional Bravais lattices we demonstrate a wide variety of spectrum transformation scenarios upon alterations in the quantum dot arrangement. This feature offers unprecedented control over the supercrystal's electromagnetic properties and enables the development of new nanophotonics materials and devices.

Multiple-Beam Surface Plasmon Holographic Nanolithography

Multiple-beam surface plasmon holographic nanolithography based on a hemispherical prism coupling configuration is numerically studied in this paper. The proposed configuration is systematically analyzed and optimized for the purpose of achieving five different two-dimensional Bravais lattices by means of adjusting the spatial distribution of three incident beams properly. Furthermore, it is shown that the variation in the relative phase between incident beams can give rise to rich periodic patterns when overlapping more than three beams for surface plasmon holographic nanolithography.

Observation of Sub-Bragg Diffraction of Waves in Crystals

We investigate the diffraction conditions and associated formation of stop gaps for waves in crystals with different Bravais lattices. We identify a prominent stop gap in high-symmetry directions that occurs at a frequency below the ubiquitous first-order Bragg condition. This sub-Bragg-diffraction condition is demonstrated by reflectance spectroscopy on two-dimensional photonic crystals with a centered rectangular lattice, revealing prominent diffraction peaks for both the sub-Bragg and first-order Bragg conditions. These results have implications for wave propagation in 2 of the 5 two-dimensional Bravais lattices and 7 out of 14 three-dimensional Bravais lattices, such as centered rectangular, triangular, hexagonal, and body-centered cubic.

Fabrication of 2D Bravais Nano Pattern and Growth of ZnO Nano Rods with Photonic Crystal Effect

Two-dimensional (2D) nano patterns including a two-dimensional Bravais lattice were fabricated by laser interference lithography using a two step exposure process. After the first exposure, the substrate itself was rotated by a certain angle, for a square or rectangular lattice, for an oblique lattice, and for a hexagonal lattice, and the and laser incident angle changed for rectangular and the and laser incident angle changed for a centered rectangular; we then carried out a second exposure process to form 2D bravais lattices. The band structure of five different 2D nano patterns was simulated by a beam propagation program. The presence of the band-gap effect was shown in an oblique and hexagonal structure. The oblique latticed ZnO nano-photonic crystal array had a pseudo-bandgap at a frequency of 0.337-0.375, 0.575-0.596 and 0.858-0.870. The hexagonal latticed ZnO nano-crystallite array had a pseudo-bandgap at a frequency of 0.335-0.384 and 0.585-0.645. The ZnO nano structure with an oblique and hexagonal structure was grown through the patterned opening window area by a hydrothermal method. The morphology of 2D nano patterns and ZnO nano structures were investigated by atomic force microscopy and scanning electron microscopy. The diameter of the opening window was approximately 250 nm. The height and width of ZnO nano-photonic crystals were 380 nm and 250 nm, respectively.

Controllable optical multi-well trap and its optical lattices using compounded cosine patterns

This paper proposes a flexible scheme to form various optical multi-well traps for cold atoms or molecules by using a simple optical system composed of an compounded amplitude cosine-only grating and a single lens illuminated by a plane light wave or a Gaussian beam. Dynamic manipulation and evolution of multi-well trap can be easily implemented by controlling the modulation frequency of the cosine patterns. It also discusses how to expand this multi-well trap to two-dimensional lattices with single- or multi-well trap by using an orthogonally or non-orthogonally modulated grating, as well as using incoherent multi-beam illumination, and these results show that all the symmetric structures of two-dimensional Bravais lattices can be obtained facilely by using proposed scheme.

Spontaneous Symmetry-Breaking Vortex Lattice Transitions in Pure Niobium

We report an extensive investigation of magnetic vortex lattice (VL) structures in single crystals of pure niobium with the magnetic field applied parallel to a fourfold symmetry axis, so as to induce frustration between the cubic crystal symmetry and hexagonal VL coordination expected in an isotropic situation. We observe new VL structures and phase transitions; all the VL phases observed (including those with an exactly square unit cell) spontaneously break some crystal symmetry. One phase even has the lowest possible symmetry of a two-dimensional Bravais lattice. This is quite unlike the situation in high-Tc or borocarbide superconductors, where VL structures orient along particular directions of high crystal symmetry. The causes of this behavior are discussed.

X-ray diffraction from an atomic plane

Reflection and transmission coefficients of X-rays by a single atomic plane are obtained in the general case where the plane consists of any two-dimensional Bravais lattice and the incident and exit X-ray beams take any direction with respect to the plane. A formula obtained for the coefficients is written in a simple form, different from that obtained by Durbin [Acta Cryst. (1995), A51, 258-268]. This makes it possible to extend Darwin's dynamical theory of X-ray diffraction to general geometries which include the cases of asymmetric skew reflection and noncoplanar multibeam diffraction.

Coulomb interactions in two-dimensional lattice structures.

Using the Van der Hoff--Benson method, general relations are derived for electric potentials and dipole-dipole interaction tensors of two-dimensional lattice systems which consist of charges or dipole moments changing periodically from site to site. Numerical values of the potentials and dipole-dipole interaction tensors as well as their asymptotics near the symmetric points of the first Brillouin zone are presented for square, triangular, and hexagonal (honeycomb) lattices. In particular, an asymptotic responsible for the formation of an incommensurate phase in triangular antiferromagnets is given. Dispersion laws for collective excitations of these lattices are calculated and compared to those derived in the nearest-neighbor approximation. The symmetry properties and lattice-sublattice relations are determined, which enables us to minimize the number of independently calculable lattice sums. For a two-dimensional Bravais lattice constituted by similar charges, at a constant unit-cell area, a dependence of the Coulomb energy on geometrical parameters of a lattice is derived. It is shown that with parameters of a triangular lattice, this dependence reaches the minimum which specifies all coefficients of long-wavelength quadratic asymptotics of the dipole-dipole interaction tensor. The low-temperature states are considered for a triangular lattice of dipoles with their orientations degenerate in the lattice plane. The temperature of the orientational phase transition is estimated by various methods and its significant characteristics are discussed. \textcopyright{} 1996 The American Physical Society.

Perturbation of the surface density of electronic states by point defects: Connection with isoenergetic curve topology

The discrete Schrodinger equation on a two-dimensional Bravais lattice perturbed by a point potential is considered. The asymptotics of perturbation of the surface density of states is investigated. It is established that the qualitative behavior of this perturbation depends mainly on the topological properties of the isoenergetic curve in the Brillouin zone.

Out of Plane Propagation of Electromagnetic Waves in a Two-dimensional Periodic Dielectric Medium

Abstract By the use of the plane wave method we have calculated the dispersion curves (photonic band structure) of electromagnetic waves propagating in a structure consisting of an infinite array of parallel, infinitely long, dielectric rods of arbitrary cross-section, characterized by a dielectric constant εa, embedded in a medium characterized by a dielectric constant εb, when the intersections of the axes of the rods with a perpendicular plane form a two-dimensional Bravais lattice. In contrast with earlier calculations of the photonic band structures of two-dimensional, periodic, dielectric structures, in the present work the electromagnetic waves are assumed to propagate out of the plane perpendicular to the rods. In numerical calculations we study a triangular lattice of air cylinders in a dielectric medium, which has recently been shown to possess a band gap common to waves of both E and H polarization for propagation in the plane perpendicular to the rods. The shifts of the edges of this absolute band gap as k 3, the component of the wave-vector of the electromagnetic waves parallel to the rods, is increased from zero are studied, as is the closing up of this gap. A new absolute band gap is found to open up below the first band as k 3 is increased. Conclusions concerning the filtering behaviour of two-dimensional, periodic, dielectric structures in the case of out-of-plane propagation of electromagnetic waves through them are drawn.

Electrostatic interactions in slabs of polarizable particles

We present results of a study of electrostatic interactions in slabs consisting of L layers with polarizable spheres on a two-dimensional Bravais lattice in an externally applied electric field. A multipole expansion to describe N-particle electrostatic interactions, introduced by the authors in a simultaneously published paper, has been used to calculate the induced dipoles on particles in the respective layers of the slab. The conclusion is that the induced dipoles differ from their bulk values if situated near the surface, and these differences depend strongly on the internal structure of the slabs. With these results the surface energy has been calculated. Moreover, the local electric field has been studied as a function of the distance from the surface of the slab. These results are compared with a mean field approximation where each particle is considered to be surrounded by a homogeneous polarization. The differences between both models are striking.

Functions of two-dimensional Bravais lattices

The symmetries of a function are considered that takes as its argument an arbitrary two-dimensional Bravais lattice, represented by a 2×2 matrix. Rotating the lattice corresponds to multiplying its matrix on the right by an element of SO(2). Matrices related by left multiplication of an element of SL(2,Z) also refer to the same lattice. Functions of matrices having these two symmetries possess an expansion known as the Roelcke–Selberg decomposition. This decomposition characterizes the function in terms of a set of coefficients (the discrete spectrum) and a function defined on a line in the complex plane (the continuous spectrum). When the function of the lattice refers to its energy, the continuous spectrum can be related to an interatomic potential between the atoms of the lattice. The decomposition can be considered a Landau theory of a strained two-dimensional Bravais lattice, valid for arbitrarily large strains, expanded about any structure. This type of theory may be useful in determining the energies of a material for many different lattice configurations from a knowledge of the energy at only a few of them.

Oreintation vibrations in two-dimensional dipole systems

A general theory is developed for low temperature orientations vibrations of dipoles on arbitrary two-dimensional Bravais lattices with any anisotropic interactions. The theory involves a method for calculating the dipole system ground state, as well as the analysis of its properties and stability (i.e. long-range order) with respect to the relatively longwave angular fluctuations. Specific results are obtained for the dipole-dipole potential. In particular, dipole orientation vibrations about the stable ferroelectric ground state on a trigonal lattice are shown to display a non-trivial asymptotic behaviour of frequency distribution g(ω) at low frequencies and specific heat c(T):g(ω)∼ω 3/2 ; c(T)∼T 5/2

On entropical structure measures in the five two‐dimensional bravais lattices

Crystal Research and TechnologyVolume 25, Issue 6 p. K146-K149 Short Note On entropical structure measures in the five two-dimensional bravais lattices Dr. D. Leuschner, Dr. D. Leuschner Anton-Graff-Str. 14, Dresden, DDR - 8019Search for more papers by this author Dr. D. Leuschner, Dr. D. Leuschner Anton-Graff-Str. 14, Dresden, DDR - 8019Search for more papers by this author First published: June 1990 https://doi.org/10.1002/crat.2170250627Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume25, Issue6June 1990Pages K146-K149 RelatedInformation

Jordan-Wigner transformation for quantum-spin systems in two dimensions and fractional statistics.

I construct a Jordan-Wigner transformation for spin-one-half quantum systems on two-dimensional lattices. I show that the spin-one-half XY (i.e., a hard-core Bose system) is equivalent (on any two-dimensional Bravais lattice) to a system of spinless fermions and gauge fields satisfying the constraint that the gauge flux on a plaquette must be proportional to the spin (particle) density on site. The constraint is enforced by the addition of a Chern-Simons term of strength \ensuremath{\theta} to the Lagrangian of the theory. For the particular value \ensuremath{\theta}=1/2\ensuremath{\pi}, the resulting particles are fermions. In general they are anyons. The implications of these results for quantum spin liquids are briefly discussed.

Translational invariance in critical phenomena Ising model on a quasi-lattice

The Ising model on a two-dimensional quasi-crystal (the Penrose tiling) is studied. Using the correlation inequality and the duality transformation bounds for the critical temperature are obtained as 1.82<Tc<7. Monte Carlo simulations on finite lattices give an estimate of Tc=2.41+or-0.02. Finite size scaling analysis of the Monte Carlo data shows that the system belongs to the same universality class as the Ising model on the two-dimensional Bravais lattices. However, the finite size scaling forms do not reproduce the asymptotic limits in the range studied whereas, in the same range, the periodic lattices are known to behave as expected.

Some static and dynamical properties of a two-dimensional Wigner crystal

The static ground-state energy of a two-dimensional Wigner crystal has been obtained for each of the five two-dimensional Bravais lattices. At constant electron number density the hexagonal lattice has the lowest energy. Phonon dispersion curves have been calculated for wave vectors along the symmetry directions in the first Brillouin zone for the hexagonal lattice. In the long-wavelength limit one of the two branches of the dispersion relation vanishes with vanishing two-dimensional wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ as $q$, the second as ${q}^{\frac{1}{2}}$. The coefficient of $q$ in the former branch is pure imaginary for certain directions of propagation in the square lattice, implying a dynamical instability of this lattice; the hexagonal lattice is stable. The vibrational zero-point energy and low-temperature thermodynamic functions have been obtained for the hexagonal lattice. The dielectric susceptibility tensor of a two-dimensional Wigner crystal ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ has been determined in the long-wavelength limit, in the presence of a static magnetic field perpendicular to the crystal, and the result has been used to obtain the dispersion relation for plasma oscillations in the electron crystal.