Six-fold Symmetry Research Articles

Realizing symmetry-guaranteed pairs of bound states in the continuum in metasurfaces

Bound states in the continuum (BICs) have received significant attention for their ability to enhance light-matter interactions across a wide range of systems, including lasers, sensors, and frequency mixers. However, many applications require degenerate or nearly degenerate high-quality factor (Q) modes, such as spontaneous parametric down conversion, non-linear four-wave mixing, and intra-cavity difference frequency mixing for terahertz generation. Previously, degenerate pairs of bound states in the continuum (BICs) have been created by fine-tuning the structure to engineer the degeneracy, yielding BICs that respond unpredictably to structure imperfections and material variations. Instead, using a group theoretic approach, we present a design paradigm based on six-fold rotational symmetry (C6) for creating degenerate pairs of symmetry-protected BICs, whose frequency splitting and Q-factors can be independently and predictably controlled, yielding a complete design phase space. Using a combination of resonator and lattice deformations in silicon metasurfaces, we experimentally demonstrate the ability to tune mode spacing from 2 nm to 110 nm while simultaneously controlling Q-factor.

En Route to High-Density Chiral Single-walled Carbon Nanotube Arrays using Solid Trojan Catalysts.

Solid catalyst is widely recognized as an effective strategy to control the chirality of single-walled carbon nanotubes (SWNTs). However, it is still not compatible with high density in horizontal arrays. "Trojan" catalysts strategy is one of the most effective methods to realize SWNTs with high density and has great potential in chirality control. Here, the co-realization of high density and chirality controlling for SWNTs in a low-temperature growth process is reported based on the developed solid "Trojan" catalyst. High temperature "Trojan" catalyst formation process provides sufficient catalyst number to acquire high density. These liquid "Trojan" catalysts are cooled to solid state by adopting low growth temperature (540°C), which can be good template to realize the chirality controlling of SWNTs with exposing six-fold symmetry face, (111). Finally, (9, 6) and (13, 1) SWNTs enriched horizontal array with the purity of ≈90% and density of 4tubesµm-1 is realized. The comparison between the distribution of initial catalysts and the density of as-grown tubes indicates no sacrificing on catalysts number to improve chirality selectivity. This work opens a new avenue on the catalyst's design and chirality controlling in SWNTs growth.

Emergent power-law interactions in near-crystalline membranes.

We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydisperse soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show that this interaction energy displays a disorder-averaged power-law behavior 〈δE〉∼Δ^{-4} at large distances Δ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy-minimized configurations of polydisperse soft disks in two dimensions.

Enabling robust anionic redox structure via tuning the symmetry of locally ordered lattice in Li-rich Li-Mn-O cathodes

Li-rich Li-Mn-O (LR-LMO) oxides are promising cathodes for low-lost and high-energy–density Li-ion batteries owing to their accessible anionic redox reaction (ARR) chemistry. However, the unstable ARR structure can result in inferior electrochemical properties, restricting practical applications of LR-LMO oxides. In this study, we demonstrate that the symmetry of locally ordered lattice can be tuned to enable robust ARR structure in LR-LMO cathodes. By employing a twofold symmetry locally ordered lattice in the LR-LMO oxide (denoted as tLR-LMO), a reversible capacity of 260 mAh/g at 0.1C and capacity retention of 78 % after 100 cycles at 1C were observed. The tLR-LMO has retarded cationic migrations during the ARR process when compared to this with a sixfold symmetry locally ordered lattice (denoted as sLR-LMO), as indicated by in-situ X-ray diffraction and ex-situ scanning transmission electron microscopy studies. The tLR-LMO also shows superior stability in its ARR structure as revealed by ex-situ X-ray absorption spectra and neutron pair distribution function results and supported by theoretical calculations. More broadly, this work highlights the role of local-structure symmetry in designing high-performance Li-rich cathodes with ARR activity.

Nonisothermal cellular automata simulation of two-dimensional snow crystal growth.

Natural snowflakes exhibit remarkable diversity with a fascinating sixfold symmetry and fractal structure. Efforts to discover the snow crystal growth pattern and mechanism behind it have been made in the past 50 years. Among those, cellular automata (CA) have made certain progress. To investigate the influence of latent heat release in snow crystal growth and have a deeper understanding of crystal growth characteristics, an improved CA model combined with vapor diffusion and heat diffusion is proposed to investigate the snowflake growth process in a two-dimensional and nonisothermal field in this paper. Simulation results show that obtained snow crystal growth behavior is consistent with previous experimental observations and satisfies the crystal growth dynamic theory. Latent heat release impacts the small structure of snow crystals, and it could have a more significant influence on snowflake morphology when vapor diffusion is slow and heat diffusion is dominating.

Three-state nematicity and magneto-optical Kerr effect in the charge density waves in kagome superconductors

The kagome lattice provides a fascinating playground to study geometrical frustration, topology and strong correlations. The newly discovered kagome metals AV$_3$Sb$_5$ (A=K, Rb, Cs) exhibit various interesting phenomena including topological band structure and superconductivity. Nevertheless, the nature of the symmetry breaking in the CDW phase is not yet clear, despite the fact that it is crucial to understand whether the superconductivity is unconventional. In this work, we perform scanning birefringence microscopy and find that six-fold rotation symmetry is broken at the onset of the CDW transition temperature in all three compounds. Spatial imaging and angle dependence of the birefringence show a universal three nematic domains that are 120$^\circ$ to each other. We propose staggered CDW orders with a relative $\pi$ phase shift between layers as a possibility to explain the three-state nematicity in AV$_3$Sb$_5$. We also perform magneto-optical Kerr effect and circular dichroism measurements on all three compounds, and the onset of the both signals is at the CDW transition temperature, indicating broken time-reversal symmetry and the existence of the long-sought loop currents in the CDW phase. Our work strongly constrains the nature of the CDWs and sheds light on possible unconventional superconductivity in AV$_3$Sb$_5$.

Rotationally Symmetrical Spoof-Plasmon Antenna for Polarization-Independent Radiation Enhancement

Plasmon antennas allow subwavelength confinement and enhancement of electromagnetic fields at the ``hotspot'' where the radiation efficiency of emitters can be substantially enhanced. Such enhancement, however, is often polarization dependent. Consequently, the radiation behaviors (e.g., radiation pattern and polarization states) of the emitter placed at the hotspot are also modified significantly. Enhancing the radiation efficiency without altering the original radiation pattern and polarization state of the emitter is highly desired for many sought-after applications involving chiral emitters but remains a challenging task, especially at low frequencies. To this end, spoof-plasmon antennas with fourfold and sixfold rotational symmetries are designed and realized experimentally. These plasmon antennas support polarization-independent localized plasmon resonances, which can significantly enhance the local density of photonic states at the structural center without changing the polarization state of the emitter. As a typical example, the structure with sixfold rotational symmetry is coupled with a half-wave dipole antenna. The measurement results show that the far-field radiation pattern of the dipole antenna is maintained with the radiation efficiency enhanced by more than 2 orders of magnitude, irrespective of the dipole orientation.

Zr3Mn3Sn4Ga: a new hexagonal Ti6Sn5-type quaternary intermetallic

The discovery of a new quaternary intermetallic compound of composition Zr3Mn3Sn4Ga during a survey of the Zr–Mn–Sn–Ga phase diagram is reported. Single-crystal X-ray diffraction reveals that the new compound crystallizes in a hexagonal lattice (space group P63/mmc, No. 194). The analysis of single-crystal X-ray diffraction data indicates that the atomic arrangement is an ordered variant of the high-temperature hexagonal Ti6Sn5 crystal structure. From a simple geometry point of view, the unit cell consists of GaZr3 octahedra chains and Mn chains growing along the sixfold symmetry axis c. The Mn chains are stacked within the ab basal plane in a kagome geometry, with large interatomic distances. The possible slight levels of Sn/Ga and Zr/Mn mixed occupancy based on additional results of electron probe microanalysis and single-crystal neutron diffraction are discussed. It is noteworthy that this compound is the first quaternary intermetallic discovered in this structure and the first Ti6Sn5 derivative bearing a magnetic metal, namely Mn, in the 6g position, which may give rise to interesting physical properties.

Search for long-range magnetic order in quasicrystals

Classification of solids is based upon the order and rotational symmetry of their atomic arrangements. Until 1984, solids were either crystalline or amorphous. In crystalline compounds, atoms or atomic clusters are ordered periodically (in a repeating pattern), and the rotational symmetries of such compounds are restricted to two-, three-, four-, and six-fold symmetry axes. Thus, five-fold or 10-fold symmetry axes, for instance, are forbidden. In amorphous compounds, the atomic arrangements are essentially disordered with no precise rotational symmetry. Here Zbigniew M. Stadnik, Professor at the University of Ottawa in the Department of Physics, looks at the magnetic order in quasicrystals.

Probing magnetic anisotropy in Kagome antiferromagnetic Mn[formula omitted]Ge with torque magnetometry

We investigate the magnetic symmetry of the topological antiferromagnetic material Mn3Ge by torque measurements. Below the Néel temperature, detailed angle-dependent torque measurements were performed on Mn3Ge single crystals in directions parallel and perpendicular to the Kagome basal plane. The out-of plane torque data exhibit ±sinθ and sin2θ behaviors, of which the former results from the spontaneous ferromagnetism within the basal plane and the latter from the in- and out-of-plane susceptibility anisotropy. The reversible component of the in-plane torque exhibits sin6φ behavior, revealing the six-fold symmetry of the in-plane magnetic free energy. Moreover, we find that the free energy minima are pinned to the direction of spontaneous ferromagnetism, which corresponds to the maxima of the irreversible component of the in-plane torque. We provide an effective spin model to describe the in-plane magnetic anisotropy. Our results demonstrate that the ground state of Mn3Ge is described by the coexistence of a strong six-fold antichiral order and a weak ferromagnetic order induced by second-order spin anisotropy.

Exploiting anisotropic particle shape to electrostatically assemble colloidal molecules with high yield and purity

HypothesisColloidal molecules with anisotropic shapes and interactions are powerful model systems for deciphering the behavior of real molecules and building units for creating materials with designed properties. While many strategies for their assembly have been developed, they typically yield a broad distribution or are limited to a specific type. We hypothesize that the shape and relative sizes of colloidal particles can be exploited to efficiently direct their assembly into colloidal molecules of desired valence. ExperimentsWe exploit electrostatic self-assembly of negatively charged spheres made from either polystyrene or silica onto positively charged hematite cubes. We thoroughly analyze the role of the shape and size ratio of particles on the cluster size and yield of colloidal molecules. FindingsUsing a combination of experiments and simulations, we demonstrate that cubic particle shape is crucial to generate high yields of distinct colloidal molecules over a wide variety of size ratios. We find that electrostatic repulsion between the satellite spheres is important to leverage the templating effect of the cubes, leading the spheres to preferentially assemble on the facets rather than the edges and corners of the cube. The sixfold symmetry of cubes favors the assembly of molecules with six, four, and two satellite spheres at appropriate size ratios and interaction strength.Furthermore, we reveal that our protocol is not affected by the specific choice of the material of the colloidal particles. Finally, we show that the permanent magnetic dipole moment of the hematite cubes can be utilized to separate colloidal molecules from non-assembled satellite particles. Our simple and effective strategy might be extended to other templating particle shapes, thereby greatly expanding the library of colloidal molecules that can be achieved with high yield and purity.

Modeling on transient microstructure evolution of solid-air solidification process under continuous cooling in liquid hydrogen

The safety hazard brought by the oxygen-rich solid formed by the solidification of air in liquid hydrogen cannot be ignored. A numerical model describing the evolution of dendrite during solidification of nitrogen-oxygen binary solutions in liquid hydrogen is developed. By introducing the reduction factor, the growth anisotropy of the six-fold symmetric dendrite simulated by the Cartesian grid is effectively reduced. The reliability of the model is verified by comparing the tip growth rate of dendrites with six-fold symmetry with analytical models. On this basis, the microstructure evolution and solute segregation of solid-air dendrites are investigated. The results show that with the increase of the cooling rate, the dendrite growth rate accelerates while aggravating the solute segregation, and the outer edge of the dendrite is more likely to reach the oxygen-rich state. The improvement in solute diffusibility enables dendrites to reach an oxygen-rich state at larger sizes, but also accelerates dendrite growth. The initial composition has little effect on the microstructure evolution and growth rate of the solid-air dendrite. However, in the presence of forced convection, the solid-air dendrite morphology loses its symmetry and makes the upstream dendrite arms reach an oxygen-rich state at a smaller size. This study helps to understand the air solidification process in liquid hydrogen and provides theoretical guidance for the safety research of liquid hydrogen system.

Quasiperiodic light.

Quasiperiodicity is a form of spatial order that has been observed in quasicrystalline matter but not light. We construct a quasicrystalline surface out of a light emitting diode. Using a nanoscale waveguide as a microscope (NSOM), we directly image the light field at the surface of the diode. Here we show, using reciprocal space representations of the images, that the light field is quasiperiodic. We explain the structure of the light field with wave superposition. Periodic ordering is limited to at most six-fold symmetry. The light field exhibits 12-fold quasisymmetry, showing order while disproving periodicity. This demonstrates that a new class, consisting of projections from hyperspace, exists in the taxonomy of light ordering.

Numerical Analysis of Dislocation Density of c ‐, a ‐, m ‐axes Grown Al 2 O 3 Crystals during Annealing Process

A numerical study of dislocation density is carried out by 3D and time-dependent analysis using the Alexander–Haasen model to estimate growth direction dependence of plastic deformation of Al2O3 single crystals during the annealing process. The direction of crystal growth is set to c-, a-, and m-axes. The calculated results show distribution of dislocation density in the crystal grown in the c-axis has a sixfold symmetry while those in the crystals grown in the a- and m-axes show asymmetric distribution. Dislocation densities in the a- and m-axes Al2O3 crystals are larger than that in the c-axis crystal.

Temperature-driven reorganization of electronic order in CsV3Sb5

We report x-ray diffraction studies of the electronic ordering instabilities in the kagome material ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ as a function of temperature and applied magnetic field. Our zero-field measurements between 10 and 120 K reveal an unexpected reorganization of the three-dimensional electronic order in the bulk of ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$: At low temperatures, a $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ superstructure modulation due to electronic order is observed, which upon warming changes to a $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}4$ superstructure at 60 K. The electronic order-order transition discovered here involves a change in the stacking of electronically ordered ${\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ layers, which coincides with anomalies previously observed in magnetotransport measurements. This implies that the temperature-dependent three-dimensional electronic order plays a decisive role for transport properties, which are related to the Berry curvature of the V bands. We also show that the bulk electronic order in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ breaks the sixfold rotational symmetry of the underlying $P6/mmm$ lattice and perform a crystallographic analysis of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ phase. The latter yields two possible superlattices, namely a staggered star-of-David and a staggered inverse star-of-David structure. Applied magnetic fields up to 10 T have no effect on the x-ray diffraction signal. This, however, does not rule out time-reversal symmetry breaking in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$.

Observation of an anisotropic ultrafast spin relaxation process in large-area WTe2 films

Weyl semimetal Td-WTe2 hosts the natural broken inversion symmetry and strong spin–orbit coupling, which contains profound spin-related physics within a picosecond timescale. However, the comprehensive understanding of ultrafast spin behaviors in WTe2 is lacking due to its limited quality of large-scale films. Here, we report on an anisotropic ultrafast spin dynamics in highly oriented Td-WTe2 films using a femtosecond pump–probe technique at room temperature. A transient spin polarization-flip transition as fast as 0.8 ps is observed upon photoexcitation. The inversed spin is subsequently scattered by defects with a duration of about 5.9 ps. The whole relaxation process exhibits an intriguing dual anisotropy of sixfold and twofold symmetries, which stems from the energy band anisotropy of the WTe2 crystalline structure and the matrix element effect, respectively. Our work enriches the insights into the ultrafast opto-spintronics in topological Weyl semimetals.

Bioinspired Co‐Design of Tactile Sensor and Deep Learning Algorithm for Human–Robot Interaction

Robots equipped with bionic skins for enhancing the robot perception capability are increasingly deployed in wide applications ranging from healthcare to industry. Artificial intelligence algorithms that can provide bionic skins with efficient signal processing functions further accelerate the development of this trend. Inspired by the somatosensory processing hierarchy of humans, the bioinspired co‐design of a tactile sensor and a deep learning‐based algorithm is proposed herein, simplifying the sensor structure while providing computation‐enhanced tactile sensing performance. The soft piezoresistive sensor, based on the carbon black‐coated polyurethane sponge, offers a continuous sensing area. By utilizing a customized deep neural network (DNN), it can detect external tactile stimulus spatially continuously. Besides, a novel data augmentation method is developed based on the sensor's hexagonal structure that has a sixfold rotation symmetry. It can significantly enhance the generalization ability of the DNN model by enriching the collected training data with generated pseudo‐data. The functionality of the sensor and the robustness of the proposed data augmentation strategy are verified by precisely recognizing five touch modalities, illustrating a well‐generalized performance, and providing a promising application prospect in human–robot interaction.

Observation of symmetry-protected corner states in breathing honeycomb topolectrical circuits

We report the experimental observation of second-order corner states in a two-dimensional breathing honeycomb topolectrical circuit with sixfold rotational symmetry C6 through voltage measurements. The topological corner states originate from the nontrivial bulk topology, which can be characterized by the topological invariant associated with the rotation eigenspectrum. We confirm two types of corner states, both originate from the C6 symmetry, while one of them is specially pinned to zero admittance because of the emerging chiral symmetry protection. We then examine the robustness of zero modes in the presence of next-nearest-neighbor hopping terms that destroy chiral symmetry but still preserve C6 symmetry. Our work provides a paradigm in circuit systems to study the exotic topological physics.

Systematic Investigation of Growth and Properties of Ga2O3 Films on C-Plane Sapphire Substrates Prepared by Plasma-Assisted Molecular Beam Epitaxy

We investigated the growth and properties of Ga2O3 films on c-plane sapphire substrates by plasma-assisted molecular beam epitaxy (PAMBE). Growth parameters generally affect the structure and properties of films. In growing Ga2O3 films on sapphire substrates by PAMBE, source flux and growth temperature are main key factors and effects of those on growth behaviors, surface morphology, crystallinity, structural quality, and optical properties have been investigated by systematic experimental design using various characterization methods including reflection high energy electron diffraction, atomic force microscope, X-ray diffraction, transmission electron microscopy etc. The source flux strongly affected growth rate of Ga2O3 films by PAMBE, and the growth temperature strongly affected surface morphology and structural properties. The orientation relationship was determined to be β-Ga2O3 //Al2O3(0001) and β-Ga2O3[010]//Al2O3[]. However, the β-Ga2O3 films were not single crystalline but showed six-fold in-plane symmetry with six rotational domain structures. The detailed rotational domain structures were investigated by transmission electron diffraction analysis. The smooth surface with the RMS value of 0.95 nm can be obtained at growth temperature of 800 °C. All the grown β-Ga2O3 films are highly transparent and the optical bandgaps were 4.79–4.89 eV.

Образование и рост фуллеренов и нанотрубок, имеющих Т-симметрию третьего порядка

According to the periodic system of fullerenes, all the fullerenes can be classified into the groups having different symmetry. It is supposed that the fullerenes of one and the same symmetry have similar properties. Before the appearance of the periodic system in 2017 the fullerenes were chosen for study at a random way that instead of ordering the results only increased information entropy. We have studied possible ways of generation and growing the fullerenes, which refer to the group having three-fold T-symmetry. Beginning with cyclopropane C3H6 producing clusters C6, we have obtained elementary fullerenes C6 as well as mini-fullerenes C12, which in their turn have produced the fullerenes from C18 to C48, perfect and imperfect, as well as nanotubes. The basic perfect fullerenes C18, C24, C30, C36, C42 and C48 have the ordinary three-fold symmetry, the intermediate ones having no such symmetry. Their imperfection is connected with extra ‘interstitial’ or carbon dimers, the dimers playing the role of defects. One can define the imperfect fullerenes with defects as the fullerenes having topological three-fold symmetry. We have calculated their shape and energies using Avogadro package and discussed possible reasons of their dependence on a fullerene size and shape. We have found that the fullerenes can be divided into two groups, alive that can grow, and dead which are impotent. Taking into account the results obtained early, allows us to make predictions that the dead fullerenes C24R, C32R, C40R and C48R of three-, four-, five- and six-fold symmetry have the most chance to be found experimentally with comparison of their isomers.