Two-dimensional Bravais Lattices Research Articles

High topological charge lasing in quasicrystals.

Photonic modes exhibiting a polarization winding akin to a vortex possess an integer topological charge. Lasing with topological charge 1 or 2 can be realized in periodic lattices of up to six-fold rotational symmetry-higher order charges require symmetries not compatible with any two-dimensional Bravais lattice. Here, we experimentally demonstrate lasing with topological charges as high as -5, +7, -17 and +19 in quasicrystals. We discover rich ordered structures of increasing topological charges in the reciprocal space. Our quasicrystal design utilizes group theory in determining electromagnetic field nodes, where lossy plasmonic nanoparticles are positioned to maximize gain. Our results open a new path for fundamental studies of higher-order topological defects, coherent light beams of high topological charge, and realizations of omni-directional, flat-band-like lasing.

Non-close-packed plasmonic Bravais lattices through a fluid interface-assisted colloidal assembly and transfer process

AbstractThe assembly of colloids at fluid interfaces followed by their transfer to solid substrates represents a robust bottom-up strategy for creating colloidal monolayers over large, macroscopic areas. In this study, we showcase how subtle adjustments in the transfer process, such as varying the contact angle of the substrate and controlling deposition speed and direction, enable the realization of all five two-dimensional Bravais lattices. Leveraging plasmonic core–shell microgels as the building blocks, we successfully engineered non-close-packed plasmonic lattices exhibiting hexagonal, square, rectangular, centered rectangular, and oblique symmetries. Beyond characterizing the monolayer structures and their long-range order, we employed extinction spectroscopy alongside finite difference time domain simulations to comprehensively investigate and interpret the plasmonic response of these monolayers. Additionally, we probed the influence of the refractive index environment on the plasmonic properties by two methods: first, by plasma treatment to remove the microgel shells, and second, by overcoating the resulting gold nanoparticle lattices with a homogeneous refractive index polymer film. Graphical Abstract

Realization of all two-dimensional Bravais lattices with metasurface-based interference lithography

Abstract Proximity-field nanopatterning (PnP) have been used recently as a rapid, cost-effective, and large-scale fabrication method utilizing volumetric interference patterns generated by conformal phase masks. Despite the effectiveness of PnP processes, their design diversity has not been thoroughly explored. Here, we demonstrate that the possibility of generating any two-dimensional lattice with diverse motifs. By controlling the amplitude, phase, and polarization of each diffraction beam, we can implement all two-dimensional Bravais lattices in three-dimensional space. The results may provide diverse applications that require three-dimensional nanostructures from optical materials and structural materials to energy storage or conversion materials.

Multipole groups and fracton phenomena on arbitrary crystalline lattices

Multipole symmetries are of interest in multiple contexts, from the study of fracton phases, to nonergodic quantum dynamics, to the exploration of new hydrodynamic universality classes. However, prior explorations have focused on continuum systems or hypercubic lattices. In this work, we systematically explore multipole symmetries on arbitrary crystal lattices. We explain how, given a crystal structure (specified by a space group and the occupied Wyckoff positions), one may systematically construct all consistent multipole groups. We focus on two-dimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all two-dimensional Bravais lattices, and on the Kagome and breathing Kagome crystal structures to illustrate the procedure on general crystal lattices. Using Wyckoff positions, we provide an in-principle classification of all possible multipole groups in any space group. We explain how, given a valid multipole group, one may construct a consistent lattice Hamiltonian and a low-energy field theory. We then explore the physical consequences, beginning by generalizing certain results originally obtained on hypercubic lattices to arbitrary crystal structures. Next, we identify two apparently novel phenomena: an emergent, robust subsystem symmetry on the triangular lattice, and an exact multipolar symmetry on the breathing Kagome lattice that does not include conservation of charge (monopole), but instead conserves a vector charge. This makes clear that there is new physics to be found by exploring the consequences of multipolar symmetries on arbitrary lattices, and this work provides the map for the exploration thereof, as well as guiding the search for emergent multipolar symmetries and the attendant exotic phenomena in real materials based on nonhypercubic lattices.

Topological measures of order for imperfect two-dimensional Bravais lattices.

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

A simple estimation of OLED light extraction enhancement due to refractive modulation lattice diffraction

To estimate the light extraction enhancement of an organic light emitting diode due to a periodic refractive modulation lattice simulating a photonic or plasmonic lattice, a noniterative scheme was proposed based on the Wentzel–Kramers–Brillouin solution used to calculate the input/output efficiency of a refractive index grating coupler for optical circuits, treating the lattice as a two-dimensional Bravais lattice.

Prediction of topological superconductivity from type-IV, -III, -II, and -I′ nodal points induced by Rashba spin-orbit coupling

Topological superconductivity (TSC) has received great theoretical and experimental attention recently. Type-I Rashba nodal point (RNP) with isotropic band dispersions and point Fermi surface (FS) induced by the Rashba spin-orbit coupling (SOC) provides a promising route to the artificial TSC, because the inherent interspin coupling (ISC) shares identical form as the $p$-wave pairing $({k}_{x}{\ensuremath{\sigma}}_{y}\ensuremath{-}{k}_{y}{\ensuremath{\sigma}}_{x})$ exactly. Here we discuss the potential TSC of other types of RNPs with different ISC forms. By constructing a generic tight-binding model with Rashba SOC, we demonstrate type-IV, -III, -II, and -I\ensuremath{'} RNPs can be achieved on two-dimensional (2D) Bravais lattices, whose FS consists of only a hole (electron) pocket, two contacted hole (electron) pockets, contacted hole and electron pockets, and point of tangency, respectively. With the coorpration of $s$-wave pairing and Zeeman gaps, these new types of RNP will evoke TSC phases with chiral Majorana edge modes (MEMs), where the Chern number will be larger than 1 for multiple symmetry-equivalent RNPs. The Chern number can be further composited when the energies of unequivalent RNPs are equal, leading to edge-dependent MEMs. Moreover, by using first-principles calculations, we demonstrate the BiSb monolayer is an ideal platform for realizing TSC with Chern number 6 from type-II, -I, or -IV RNP. This work enriches the types of nodal point induced by Rashba SOC and offers a generic guidance on realizing multiple and edge-dependent MEMs from the abundantly synthesized 2D surface metal layers.

Geometrical Layout Effect on Light Intensity Distribution in SSI-LED

Like the light emission phenomenon in an incandescent light bulb, Kuo’s group have reported a broad-band white light emission from a new type of solid-state incandescent light emitting device (SSI-LED) (1-4). The device is a simple metal-oxide-semiconductor (MOS) capacitor with a high-k gate dielectric deposited over a p-type Si wafer (1-4). On hard breakdown of the dielectric, nano-sized conductive paths (or, nano-resistors) are formed in the high-k stack. Large passage of current through these conductive paths results in light emission due to the thermal excitation. SSI-LEDs have been shown to have very high color rendering index (CRI) of up to 97.9 (4). This indicates that the light emission process from the conductive paths is close to black body radiation phenomenon. This fact along with temperature estimates made for nano-resistors using simple energy balance-based calculations has been previously utilized to perform light intensity distribution calculations over SSI-LED. (5) In this paper, the authors adapt a similar approach and study the effect of geometrical layout of nano-resistors on the emitted light intensity distribution from SSI-LED at higher magnifications.Stefan-Boltzmann law quantitates the total energy radiated per unit surface area of a black body across all wavelengths per unit time j^* and is directly proportional to the fourth power of the black body's temperature T. If a black body of surface area A is radiating energy to the surroundings at temperature Tc, the net power radiated can be accounted as given in equationP=Aeσ(T4-Tc4) (1)where e is the emissivity of the object and σ is the Stefan–Boltzmann constant. Under the assumption that the nano-resistors formed in the dielectric act as point light sources, emitted light intensity at certain distance from the nano-resistors can be calculated by following inverse-square law. The resultant intensity from all the nano-resistors at any given point in space is simply the scalar addition of radiation intensity from each of the individual nano-resistors, i.e., if I1, I2...., IN are the radiation intensities due to N nano-resistors at a point in space then resultant intensity,Iresultant = I1+ I2+I3.........+ IN Reasonable assumptions are made for various physical parameters while performing radiation intensity distribution calculations. For example, nano-resistors with circular cross-sections were assumed to form in the high-k dielectric. Three different diameter sizes of nano-resistors are considered – 20 nm, 50 nm and 100 nm. Four different layout configurations (linear, triangular, square, and face-centered square) of nano-resistors are studied for localized light intensity distribution calculations based on the two-dimensional Bravais lattice symmetries. Intensity calculations are done for two cases. In the first case, nano-resistors of same sizes are considered in all the different geometrical layout configurations. In the second case, larger sized nano-resistors of diameter 100 nm are formed in the vicinity of smaller sized nano-resistor keeping the geometrical layout to be same. The light absorption by ITO gate in the SSI-LED is accounted using Beer-Lambert’s law.The geometrical layout of the nano-resistors formed in the SSI-LED affected the local light intensity distribution at higher magnifications and will be presented and discussed in detail. The simulated distribution could be correlated to the experimentally observed light dot patterns formed in SSI-LED at higher magnifications, as shown in figure 1 (6).1. Y. Kuo and C.-C. Lin, Appl. Phys. Lett., 102, 031117 (2013).2. Y. Kuo and C.-C. Lin, ECS Solid-State Lett., 2, Q59 (2013).3. Y. Kuo and C.-C. Lin, Solid-State Electron., 89, 120 (2013).4. C.-C. Lin and Y. Kuo, Appl. Phys. Lett., 106, 121107 (2015).5. A. Shukla and Y. Kuo, Prime 2020, #H04-1965 (2020)6. C.-C. Lin, PhD. Thesis, Texas A&M University, College Station, Texas (2014). Figure 1

Minimizing lattice structures for Morse potential energy in two and three dimensions

We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, fcc, and bcc lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from bcc, fcc to hcp, as α increases, that we partially and heuristically explain from the lattice theta function properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.

Extremal spectral gaps for periodic Schrödinger operators

The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the mth spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the mth gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.

Ordered Particle Arrays via a Langmuir Transfer Process: Access to Any Two-Dimensional Bravais Lattice.

We demonstrate how to directly transform a close-packed hexagonal colloidal monolayer into nonclose-packed particle arrays of any two-dimensional symmetry at the air/water interface. This major advancement in the field of nanoparticle self-assembly is based on a simple one-dimensional stretching step in combination with the particle array orientation. Our method goes far beyond existing strategies and allows access to all possible two-dimensional Bravais lattices. A key element of our work is the possibility to macroscopically stretch a particle array in a truly one-dimensional manner, which has not been possible up to now. We achieve this by stretching the nanoparticle array at an air/water interface during the transfer process. The degree of stretching is simply controlled by the wettability of the transfer substrate. To retain the symmetry of the transferred structure, the capillary forces upon drying have to be circumvented. We demonstrate two concepts based on thermal fixation for this. It allows for the first time to fabricate nonclose-packed, nonhexagonal colloidal monolayers on a macroscopic length scale.

2D Wigner crystals on silicon surface induced by nanosecond pulsed laser

During pulsed laser interaction on silicon surface, the transient electron lattices occur on plasma resonance induced from faster photons on surface in our experiment. Here, it is interesting that the electron crystals can be built on plasma surface induced by intensive faster photons on silicon, which is similar with Wigner crystal in intensive electric field or intensive magnetic field. In the two-dimensional (2D) Bravais lattices, the electron crystals of three-fold, four-fold and six-fold symmetry lattices can be observed on plasma surface by using the reflecting Talbot effect images. On these surface period lattices, the decay spectrum of faster emission on the 2D transient electron crystal was measured, which is related to the faster decay peak with lifetime of 100 ns. These new phenomena have a good application in preparing the PN junction of photo-electronic devices, where the surface electron gas on plasma surface can inject into the top layer to build N type region, and the positive ions under plasma can inject into the bottom layer for forming P type region on silicon with defects in vacuum.

Optimal lattice configurations for interacting spatially extended particles

We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.

Diophantine Approach to the Classification of Two-Dimensional Lattices: Surfaces of Face-Centered Cubic Materials.

The long-range periodic order of a crystalline surface is generally represented by means of a two-dimensional Bravais lattice, of which only five symmetrically distinct types are possible. Here, we explore the circumstances under which each type may or may not be found at the surfaces of face-centered cubic materials and provide means by which the type of lattice may be determined with reference only to the Miller indices of the surface; the approach achieves formal rigor by focusing on the number theory of integer variables rather than directly upon real geometry. We prove that the {100} and {111} surfaces are, respectively, the only exemplars of square and triangular lattices. For surfaces exhibiting a single mirror plane, we not only show that rectangular and rhombic lattices are the only two possibilities, but also capture their alternation in terms of the parity of the indices. In the case of chiral surfaces, oblique lattices predominate, but rectangular and rhombic cases are also possible and arise according to well-defined rules, here partially recounted.

7a-X-4 二次元Bravais格子からのX線回折

7a-X-4 二次元Bravais格子からのX線回折

Band warping, band non-parabolicity, and Dirac points in electronic and lattice structures

We illustrate at a fundamental level the physical and mathematical origins of band warping and band non-parabolicity in electronic and vibrational structures. We point out a robust presence of pairs of topologically induced Dirac points in a primitive-rectangular lattice using a p-type tight-binding approximation. We analyze two-dimensional primitive-rectangular and square Bravais lattices with implications that are expected to generalize to more complex structures. Band warping is shown to arise at the onset of a singular transition to a crystal lattice with a larger symmetry group, which allows the possibility of irreducible representations of higher dimensions, hence band degeneracy, at special symmetry points in reciprocal space. Band warping is incompatible with a multi-dimensional Taylor series expansion, whereas band non-parabolicities are associated with multi-dimensional Taylor series expansions to all orders. Still band non-parabolicities may merge into band warping at the onset of a larger symmetry group. Remarkably, while still maintaining a clear connection with that merging, band non-parabolicities may produce pairs of conical intersections at relatively low-symmetry points. Apparently, such conical intersections are robustly maintained by global topology requirements, rather than any local symmetry protection. For two p-type tight-binding bands, we find such pairs of conical intersections drifting along the edges of restricted Brillouin zones of primitive-rectangular Bravais lattices as lattice constants vary relatively to each other, until these conical intersections merge into degenerate warped bands at high-symmetry points at the onset of a square lattice. The conical intersections that we found appear to have similar topological characteristics as Dirac points extensively studied in graphene and other topological insulators, even though our conical intersections have none of the symmetry complexity and protection afforded by the latter more complex structures.

Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

Statistical topology of perturbed two-dimensional lattices

.The Voronoi cell of any atom in a lattice is identical. If atoms are perturbed from their lattice coordinates, then the topologies of the Voronoi cells of the atoms will change. We consider the distribution of Voronoi cell topologies in two-dimensional perturbed systems. These systems can be thought of as simple models of finite-temperature crystals. We give analytical results for the distribution of Voronoi topologies of points in two-dimensional Bravais lattices under infinitesimal perturbations and present a discussion with numerical results for finite perturbations.

Arbitrary lattice symmetries via block copolymer nanomeshes.

Self-assembly of block copolymers is a powerful motif for spontaneously forming well-defined nanostructures over macroscopic areas. Yet, the inherent energy minimization criteria of self-assembly give rise to a limited library of structures; diblock copolymers naturally form spheres on a cubic lattice, hexagonally packed cylinders and alternating lamellae. Here, we demonstrate multicomponent nanomeshes with any desired lattice symmetry. We exploit photothermal annealing to rapidly order and align block copolymer phases over macroscopic areas, combined with conversion of the self-assembled organic phase into inorganic replicas. Repeated photothermal processing independently aligns successive layers, providing full control of the size, symmetry and composition of the nanoscale unit cell. We construct a variety of symmetries, most of which are not natively formed by block copolymers, including squares, rhombuses, rectangles and triangles. In fact, we demonstrate all possible two-dimensional Bravais lattices. Finally, we elucidate the influence of nanostructure on the electrical and optical properties of nanomeshes.

Observation of the two-dimensional reciprocal lattice by use of lattice grating sheets and a laser pointer

Demonstration of the diffraction patterns from the two-dimensional Bravais lattice has been studied by use of the two single line lattice grating sheets and a laser pointer. A variable two-dimensional lattice grating was prepared using two grating sheets which are closely attached to each other. The five types of two-dimensional Bravais lattices can be produced by adjusting the relative angle between two single line lattices. The light diffraction patterns from the two-dimensional Bravais lattices indicate the reciprocal lattices of these basic two-dimensional lattice structures.