Six-fold Symmetry Research Articles

Visualizing symmetry-breaking electronic orders in epitaxial Kagome magnet FeSn films

Kagome lattice hosts a plethora of quantum states arising from the interplay of topology, spin-orbit coupling, and electron correlations. Here, we report symmetry-breaking electronic orders tunable by an applied magnetic field in a model Kagome magnet FeSn consisting of alternating stacks of two-dimensional Fe3Sn Kagome and Sn2 honeycomb layers. On the Fe3Sn layer terminated FeSn thin films epitaxially grown on SrTiO3(111) substrates, we observe trimerization of the Kagome lattice using scanning tunneling microscopy/spectroscopy, breaking its six-fold rotational symmetry while preserving the translational symmetry. Such a trimerized Kagome lattice shows an energy-dependent contrast reversal in dI/dV maps, which is significantly enhanced by bound states induced by Sn vacancy defects. This trimerized Kagome lattice also exhibits stripe modulations that are energy-dependent and tunable by an applied in-plane magnetic field, indicating symmetry-breaking nematicity from the entangled magnetic and charge degrees of freedom in antiferromagnet FeSn.

Design and mechanical properties of 3D circular curve transversal-isotropic auxetic structure

In this work, the 3D circular curve transversal-isotropic auxetic structure (3D-CCTAS) is designed based on a six-fold rotation symmetry in space and combines circular-curve and re-entrant hexagonal structures. Based on energy methods to obtain the expressions of Young's modulus and Poisson's ratio for 3D-CCTAS in transverse and longitudinal directions and numerical simulations by imposing periodic boundary conditions on the unit cell. Young's modulus and Poisson's ratio of 3D-CCTAS are investigated by numerical simulation and theoretical analysis to study the influence of structural parameters (angle of the circular curve section θ, rod width b, and rod thickness t). Both numerical simulation and theoretical analysis are also verified by the uniaxial compression experiment of 3D-CCTAS, for which the results show satisfactory concordance. Significantly, the transverse isotropy of 3D-CCTAS is determined by the spatial specificity of the structure, and therefore 3D-CCTAS exhibits transverse isotropy no matter how much θ, t, or b is changed.

Atomistic Understanding of the Competition between Dislocation and Twinning in Silver under Nanoindentation

In this study, the competition mechanisms between dislocation slip and twinning in silver with a low stacking fault energy using molecular dynamics (MD) simulation from an atomistic point of view are reported. Herein, three crystallographic surface orientations of , , and are considered and compared. The indentation stress–strain curves are successfully obtained from the load–displacement curves of nanoindentation. The stress of , , and orientations drops at the strains of 0.140, 0.133, and 0.136, which corresponds to the yield stress of 3.83, 4.33, and 4.99 GPa, respectively. Dislocation slip and twinning simultaneously form in silver as indicated by the total potential energy of the system. Furthermore, the typical four‐, two‐, and sixfold symmetries of the out‐of‐plane displacement as in copper are not observed for , , and orientations in silver. Hence, this observation can be supported by the simultaneous occurrence of dislocation slip and twinning in silver.

Nanostars in Highly Si-Doped GaN.

Understanding the relation between surface morphology during epitaxy of GaN:Si and its electrical properties is important from both the fundamental and application perspectives. This work evidences the formation of nanostars in highly doped GaN:Si layers with doping level ranging from 5 × 1019 to 1 × 1020 cm-3 grown by plasma-assisted molecular beam epitaxy (PAMBE). Nanostars are 50-nm-wide platelets arranged in six-fold symmetry around the [0001] axis and have different electrical properties from the surrounding layer. Nanostars are formed in highly doped GaN:Si layers due to the enhanced growth rate along the a-direction ⟨112̅0⟩. Then, the hexagonal-shaped growth spirals, typically observed in GaN grown on GaN/sapphire templates, develop distinct arms that extend in the a-direction ⟨112̅0⟩. The nanostar surface morphology is reflected in the inhomogeneity of electrical properties at the nanoscale as evidenced in this work. Complementary techniques such as electrochemical etching (ECE), atomic force microscopy (AFM), and scanning spreading resistance microscopy (SSRM) are used to link the morphology and conductivity variations across the surface. Additionally, transmission electron microscopy (TEM) studies with high spatial resolution composition mapping by energy-dispersive X-ray spectroscopy (EDX) confirmed about 10% lower incorporation of Si in the hillock arms than in the layer. However, the lower Si content in the nanostars cannot solely be responsible for the fact that they are not etched in ECE. The compensation mechanism in the nanostars observed in GaN:Si is discussed to be an additional contribution to the local decrease in conductivity at the nanoscale.

Hierarchy of anomalies in the simple rose model of water

The density, diffusion, and structural anomalies of the simple two-dimensional model of water were determined by Monte Carlo simulations. The rose model was used which is a very simple model for explaining the origin of water properties. Rose water molecules are modelled as two-dimensional Lennard-Jones disks with rose potentials for orientation dependent pairwise interactions mimicking formations of hydrogen bonds. The model can be seen also as a variance of silica-like models. Two parameters of potential in this work were selected in a way that (1) the model exhibits similar properties to Mercedes-Benz (MB) water model; and (2) that the model has real-like properties of water. Beside the known thermodynamic anomaly for the model we also found diffusion and structural anomalies. The orientational order parameters were calculated and maximum encountered for three and six-fold symmetry. For the MB parametrization, the anomalies occur in hierarchy order, which is a slight variation of the hierarchy order in real water. The diffusion anomaly region is the innermost in the hierarchy while for water it is the density anomaly region. In case of real water parametrization the most inner is the structural anomaly.

Anisotropic frictional characteristics among MoS2/SiO2 layer-dependent heterojunctions

The findings pertaining to the two-dimensional single-layer interface of MoS2 and SiO2 reveal that the matched heterostructure exhibits a prominent frictional anisotropy with a hexagonal six-fold symmetry. Notably, the misaligned and aligned contacts exhibit distinct frictional coefficients of 0.074 and 0.059, respectively. Moreover, as the number of layers increases, the features of frictional anisotropy decline considerably until reaching nine layers. The adaptive nanodevices derived from this research may hold significant potential for application in various heterojunction configurations.

Anisotropy of nanoscale friction: Influence of lattice structure, temperature, and wear

The anisotropy of nanoscale friction is analyzed for different samples using friction force microscopy under ultrahigh-vacuum conditions with varying scan directions. For both $\mathrm{Mo}{\mathrm{S}}_{2}$ and highly oriented pyrolytic graphite surfaces, our experiments reveal a sixfold symmetry of the friction vector plots in accordance with solutions of the Langevin equation, where the atomic force microscope tip is treated as a point mass elastically driven on the hexagonal surface lattices. In addition, the effect of temperature is analyzed both by experiment and by simulations. In both cases we find that increased temperature not only lowers the absolute friction values but also decreases the friction anisotropy. In contrast, experiments on NaCl cannot be explained on a similar basis. While the square symmetry is clearly revealed in the first measurements, it progressively deteriorates with repeated scanning, which is possibly due to abrasive wear occurring at the interface.

Modeling of solid-air multi-dendrite growth evolution driven by coupled thermal-solute using non-isothermal quantitative phase field method

The safety hazard posed by residual or infiltrated air condensing into external oxygen-rich solid-air (SA) particles in the liquid hydrogen container deserves attention. In this study, the patterns of morphological evolution, oxygen solute distribution, and growth rate of SA single- and multi-dendrite under various thermal environments are investigated using a non-isothermal quantitative phase field model for the first time. The lattice anisotropy problem of six-fold symmetric dendrites is effectively avoided. The results show that the Type II boundary condition is more suitable for the simulation of dendrite growth, which can avoid the influence of boundary conditions on dendrite growth. In addition, the suitability of the isotropic difference method for simulating the growth of SA multi-dendrites with six-fold symmetry is verified in this paper. The maximum values of solid fraction and oxygen solute concentration within the simulated domain are 0.62 and 0.435, respectively, for an initial subcooling degree of 4 K. The temperature distribution is significantly altered by the application of boundary heat flux and affects the growth pattern of SA dendrites. Compared to the scenario with no boundary heat flux, unilateral heat input and heat extraction decreased the solid fraction of SA dendrites by 31% and increased it by 43%, respectively, while the maximum concentration of oxygen solute was reduced to 144.8% and 97.3%, respectively. When boundary heat flux was applied to all quadrilateral boundaries, the solid fraction under heat extraction and heat input conditions is reduced by 82% and enhanced by 138%, respectively, compared to the scenario with no boundary heat flux. Additionally, the maximum concentration of oxygen solute is found to be 197.6% and 69.5% of that without boundary heat flux, respectively. This study improves the understanding of the evolution of solid-air dendrite growth and can provide theoretical guidance for the safe use of liquid hydrogen systems.

Tunability of spin-wave spectra in a 2D triangular shaped magnonic fractals

Reprogramming the structure of the magnonic bands during their operation is important for controlling spin waves in magnonic devices. Here, we report the tunability of the spin-wave spectra for a triangular shaped deterministic magnonic fractal, which is known as Sierpinski triangle by solving the Landau–Lifshitz–Gilbert equation using a micromagnetic simulations. The spin-wave dynamics change significantly with the variation of iteration number. A wide frequency gap is observed for a structure with an iteration number exceeding some value and a plenty of mini-frequency bandgaps at structures with high iteration number. The frequency gap could be controlled by varying the strength of the magnetic field. A sixfold symmetry in the frequency gap is observed with the variation of the azimuthal angle of the external magnetic field. The spatial distributions of the spin-wave modes allow to identify the bands surrounding the gap. The observations are important for the application of magnetic fractals as a reconfigurable aperiodic magnonic crystals.

Exotic liquid crystalline phases in monolayers of vertically vibrated granular particles

ABSTRACTVibrated monolayers of granular particles confined into horizontal cavities form a variety of fluid patterns with orientational order that resemble equilibrium liquid-crystal phases. In some cases, one can identify nematic and smectic patterns that can be understood in terms of classical statistical mechanics of hard bodies. Low aspect ratio cylinders project as rectangles and form uniaxial, or 2-atic, and tetratic, or 4-atic, nematic phases. Other polygonal particles may exhibit different liquid-crystal phases, in general p-atic phases, of higher symmetries. We give a brief summary of theoretical work on rectangles and triangles, and provide some experimental results on vibrated monolayers. In the case of equilateral triangles, the theory predicts an exotic triatic phase, or 6-atic phase, with six-fold symmetry and three equivalent directors. The right-angled triangles exhibit a 4-atic phase with strong octatic (8-atic) correlations. Experiments on cylinders show 4-atic textures and, even more remarkable, geometric frustration caused by confinement excites topological defects, which seem to follow the same topological rules as standard liquid crystals. Some of our findings can be understood with the help of simulations of hard particles subject to thermal equilibrium, although standard Density-Functional Theories fail to account for the correct equilibrium phases in some cases.

Surface electronic structure of Re(0001): A spin-resolved photoemission study

The surface electronic structure of Re(0001) has been investigated in a combined experimental and theoretical study. Spin- and angle-resolved photoemission was employed to unravel the spin-dependent $E({\mathbf{k}}_{\ensuremath{\parallel}})$ dispersion of electronic states along the $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\mathrm{M}}$ and $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\mathrm{K}}$ directions. The results are compared with band-structure calculations based on density-functional theory. Additional calculations include transitions into final states by taking into account the corresponding matrix elements. Recently, Shockley- and Tamm-type surface states close to ${E}_{\mathrm{F}}$ were identified, which were found to be mixed by spin-orbit coupling. Here, we determined the Rashba parameter ${\ensuremath{\alpha}}_{\mathrm{R}}$ of the surface state around $\overline{\mathrm{\ensuremath{\Gamma}}}$ to 0.32 and $0.34\phantom{\rule{4pt}{0ex}}\mathrm{eV}\phantom{\rule{0.16em}{0ex}}\stackrel{\ifmmode \mathring{}\else \r{}\fi{}}{\mathrm{A}}$ along $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\mathrm{M}}$ and $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\mathrm{K}}$, respectively. Furthermore, we extend our analysis to a wider $E({\mathbf{k}}_{\ensuremath{\parallel}})$ range revealing a multitude of electronic states along both high-symmetry directions. In particular, Rashba-type spin splittings are observed around the high-symmetry $\overline{\mathrm{\ensuremath{\Gamma}}}$ and $\overline{\mathrm{M}}$ points. At variance with theoretical predictions for a perfect hcp(0001) surface, we do not find any out-of-plane spin polarization. This is caused by monatomic steps of a real Re(0001) surface with alternating terminations, leading on average to an effective sixfold surface symmetry and vanishing net out-of-plane spin polarization.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible -wave symmetry, in place of the conventional -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Manipulating the nematic director by magnetic fields in the spin-triplet superconducting state of CuxBi2Se3

Electronic nematicity, a consequence of rotational symmetry breaking, is an emergent phenomenon in various new materials. In order to fully utilize the functions of these materials, ability of tuning them through a knob, the nematic director, is desired. Here we report a successful manipulation of the nematic director, the vector order-parameter (d-vector), in the spin-triplet superconducting state of CuxBi2Se3 by magnetic fields. At H = 0.5 T, the ac susceptibility related to the upper critical field shows a two-fold symmetry in the basal plane. At H = 1.5 T, however, the susceptibility shows a six-fold symmetry, which has never been reported before in any superconductor. These results indicate that the d-vector initially pinned to a certain direction is unlocked by a threshold field to respect the trigonal crystal symmetry. We further reveal that the superconducting gap in different crystals converges to p_x symmetry at high fields, although it differs at low fields.

Growth of Hybrid Perovskite Crystals from CH3NH3PbI3–xClx Solutions Subjected to Constant Solvent Evaporation Rates

In this work, we subjected hybrid lead-mixed halide perovskite (CH3NH3PbI3-xClx) precursor inks to different solvent evaporation rates in order to facilitate the nucleation and growth of perovskite crystals. By controlling the temperature of perovskite solutions placed within open-air rings in precise volumes, we established control over the rate of solvent evaporation and, thus, over both the growth rate and the shape of perovskite crystals. Direct utilization of diluted lead-mixed halide perovskites solutions allowed us to control the nucleation and to favor the growth of only a low number of perovskite crystals. Such crystals exhibited a clear sixfold symmetry. While crystals formed at a lower range of temperatures (40-60 °C) exhibited a more compact dendritic shape, the crystals grown at a higher temperature range (80-110 °C) displayed a fractal dendritic morphology.

Circumcoronenes.

Circumcoronene, a hexagonal graphene fragment with six zigzag edges, has been the focus of theoretical studies for many years, but its synthesis in solution has remained a challenge. In this study, we present a facile method for synthesizing three derivatives of circumcoronene using Brønsted/Lewis acid-mediated cyclization of vinyl ether or alkyne. Their structures were confirmed through X-ray crystallographic analysis. Bond length analysis, NMR measurement, and theoretical calculations showed that circumcoronene mostly follows Clar's bonding model and exhibits dominant local aromaticity. Its absorption and emission spectra are similar to those of the smaller hexagonal coronene due to its six-fold symmetry.

Self-assembled GA-Repeated Peptides as a Biomolecular Scaffold for Biosensing with MoS2 Electrochemical Transistors.

Biosensors with two-dimensional materials have gained wide interest due to their high sensitivity. Among them, single-layer MoS2 has become a new class of biosensing platform owing to its semiconducting property. Immobilization of bioprobes directly onto the MoS2 surface with chemical bonding or random physisorption has been widely studied. However, these approaches potentially cause a reduction of conductivity and sensitivity of the biosensor. In this work, we designed peptides that spontaneously align into monomolecular-thick nanostructures on electrochemical MoS2 transistors in a non-covalent fashion and act as a biomolecular scaffold for efficient biosensing. These peptides consist of repeated domains of glycine and alanine in the sequence and form self-assembled structures with sixfold symmetry templated by the lattice of MoS2. We investigated electronic interactions of self-assembled peptides with MoS2 by designing their amino acid sequence with charged amino acids at both ends. Charged amino acids in the sequence showed a correlation with the electrical properties of single-layer MoS2, where negatively charged peptides caused a shift of threshold voltage in MoS2 transistors and neutral and positively charged peptides had no significant effect on the threshold voltage. The transconductance of transistors had no decrease due to the self-assembled peptides, indicating that aligned peptides can act as a biomolecular scaffold without degrading the intrinsic electronic properties for biosensing. We also investigated the impact of peptides on the photoluminescence (PL) of single-layer MoS2 and found that the PL intensity changed sensitively depending on the amino acid sequence of peptides. Finally, we demonstrated a femtomolar-level sensitivity of biosensing using biotinylated peptides to detect streptavidin.

Planes of isotropic shear moduli in anisotropic materials

The elastic properties of crystalline solids are known to be anisotropic in general. The shear modulus G(n,m) depends upon the shear plane normal n and the shear direction m. It is known that G(n,m) is independent of m when n is normal to planes with three, four or sixfold symmetries. Additionally, G(n,m) is always independent of m when n is parallel to the <001> directions in cubic crystals, even in point groups 23 and m3‾ where they possess only a twofold symmetry. In the current study, an expression for ΔG(n), the difference between the extreme values of G(n,m) within a shear plane, is derived for a general anisotropic material. Novel 3D surfaces, with the radii along n related to ΔG(n), representing the anisotropy of the shear modulus within the shear plane, are presented. It is further shown that shear planes on which G(n,m) is independent of m can exist in all crystal systems. A general equation to determine n with isotropic G is derived. For all materials except those with monoclinic and triclinic crystal symmetries, explicit expressions for the isotropic shear planes are obtained. It is found that G(n,m) can be independent of m on planes without symmetry in addition to the ones with three, four and sixfold symmetries.

Protein nanoarrays using the annexin A5 two-dimensional crystal on supported lipid bilayers.

Protein nanoarrays are regularly ordered patterns of proteins fixed on a solid surface with a periodicity on the order of nanometers. They have significant potential applications as highly sensitive bioassays and biosensors. While several researchers have demonstrated the fabrication of protein nanoarrays with lithographic techniques and programmed DNA nanostructures, it has been difficult to fabricate a protein nanoarray containing a massive number of proteins on the surface. We now report the fabrication of nanoarrays of streptavidin molecules using a two-dimensional (2D) crystal of annexin A5 as a template on supported lipid bilayers that are widely used as cell membranes. The 2D crystal of annexin A5 has a six-fold symmetry with a period of about 18 nm. There is a hollow of a diameter of about 10 nm in the unit cell, surrounded by six trimers of annexin A5. We found that a hollow accommodates up to three streptavidin molecules with their orientation controlled, and confirmed that the molecules in the hollow maintain their specific binding capability to biotinylated molecules, which demonstrates that the fabricated nanoarray serves as an effective biosensing platform. This methodology can be directly applied to the fabrication of nanoarrays containing a massive number of any other protein molecules.

Heredity of clusters in liquid Ta rapid solidification process and its correlation with local symmetry

Metallic glass (MG) has received intensive attention in the fields of amorphous physics and materials science, owing to its excellent mechanical properties, good corrosion resistance, and large elastic deformation limit. Comparing with traditional oxide glass, the limited glass-forming ability (GFA) seriously restricts the application of MG in engineering. Therefore, the GFA has been a hot scientific issue in the field of amorphous material research. Recently, scientists have fully realized that GFA is closely related to the local atomic structure in liquid as well as its evolution features. Since the MG is called the “freezing” liquid, exploring the correlation of local atomic structures between liquid phase and solid phase under rapid solidification conditions is helpful in understanding the microstructural mechanism of GFA. Therefore, the rapid solidification process of liquid Ta is investigated via molecular dynamics simulation. The pair correlation function (PDF), the largest standard cluster (LSC), and the reverse atomic trajectory tracking methods are used to characterize and analyze the microstructure and its evolution during the rapid solicitation of liquid Ta. The results show that the local atomic configurations of the rapidly solidified Ta are various Kasper clusters as well as their distorted configurations, among of which [1/444, 10/555, 2/666] deformed icosahedron (or Z13 cluster) accounts for the highest proportion. The trend of hereditary ability of clusters revealed by the onset temperature of continuous heredity is consistent well with that by the fraction of staged heredity. The geometric symmetry of clusters can be quantitatively characterized by using the local symmetry parameter (LSP). The hereditary ability of clusters is closely related to their LSP. The local five-fold symmetry is beneficial to enhancing hereditary ability, while local four- and six-fold symmetry are disadvantageous for that. The probability of clusters with the same LSC index emerging in the energy range follows the Gaussian distribution, and the expected average atomic potential energy &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ {E}_{\rm exp}^{j} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231153_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231153_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is almost linearly related to the LSP, and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ {E}_{\rm exp}^{j} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231153_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231153_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; decreases with the increase of LSP&lt;sub&gt;5&lt;/sub&gt;. The high local five-fold symmetry reduces the average atomic potential energy of LSC, thereby enhancing its configurational heredity. These findings have guiding significance in improving GFA through regulating the local symmetry of liquid monatomic metals or alloys.

Importance of annexin V N-terminus for 2D crystal formation and quick purification protocol of recombinant annexin V.

Annexin V forms trimeric structures which further assemble into two-dimensional crystal (2D crystal) lattices on negatively charged phospholipid bilayer in a Ca2+-dependent manner. It is also known that annexin V 2D crystals show two types of symmetric patterns with six-fold symmetry (p6) and three-fold symmetry (p3). The p6 lattice also contains additional trimers in the gaps between the p6 axes, which are also referred to as non-p6 trimers because they do not participate in the formation of the p6 lattice. We here show that the annexin V N-terminal has significant influence on 2D crystal formation using high-speed atomic force microscopy (HS-AFM) observations. We also present a quick purification method to purify recombinant annexin V without any residual affinity tag after protein purification in ~3h.