Equilateral Triangular Lattice Research Articles

K2Ni(SeO3)2: A Perfect S = 1 Triangular-Lattice Antiferromagnet with Strong Geometric Frustration and Easy-Plane Anisotropy

We synthesized a single crystal of K2Ni(SeO3)2 (S = 1) with an equilateral triangular lattice (space group R-3m). The magnetic susceptibility χ(T) and heat capacity Cp(T) show that the compound is antiferromagnetically ordered at TN ≈ 7 K. The Curie–Weiss temperature is about θCW = −72.41 and −71.83 K when the field is applied along the ab plane (H//ab) and parallel to the c axis (H//c), respectively, indicating a strong geometric frustration (f ≈ 10). Both the χ(T) and M(H) point to the presence of easy-plane anisotropy. Density functional theory (DFT) calculations estimate the intraplane exchange (J/kB = −19.04 K) and the interplane exchange (J′/kB = −0.01 K), proving the excellent two-dimensionality. The 50 T high-field magnetization M(H) at 2 K indicates a sign of 1/3 magnetization plateau. The high-field electron spin resonance demonstrates the antiferromagnetic resonance modes with easy-plane anisotropy, yielding J highly consistent with the DFT result. Thus, K2Ni(SeO3)2 is a perfect triangular-lattice antiferromagnet with strong frustration and easy-plane anisotropy.

Topology optimization of the periodic pile barrier with initial stresses arranged in rectangular and equilateral triangular lattices

The attenuation zones (AZs) of periodic pile barriers can be used to suppress ambient vibration in specific frequency ranges. Topology optimization of periodic pile units can search for efficient pile structures. However, in the process of optimization, the influence of initial stresses on the AZs is usually ignored, and the periodic piles are often arranged in a square lattice. In this paper, to maximize the relative width of AZs (RWAZs), the genetic algorithm (GA) and nondominated sorting genetic algorithm-II (NSGA-II) are adopted for single- and multiobjective topology optimization of the periodic pile unit with different initial stresses. Periodic piles arranged in rectangular lattices with different aspect ratios and those in an equilateral triangular lattice are considered, and the corresponding optimized pile structures are obtained. It is found that with the same unit area, the periodic piles with the equilateral triangular layout correspond to the largest RWAZ. Due to the existence of the initial stress, the position and width of the AZs of the optimized pile structure decrease. The RWAZ of the optimized pile structure decreases with increasing initial stresses. Finally, numerical results from both frequency domain and time domain analyses demonstrate that the optimized pile barrier can effectively isolate elastic waves, which shows extensive application prospects in engineering practice.

Optical Conductivity Spectra of Charge-Crystal and Charge-Glass States in a Series of θ-Type BEDT-TTF Compounds

In the 3/4-filled band system θ-(BEDT-TTF)2X with a two-dimensional triangular lattice, charge ordering (CO) often occurs due to strong inter-site Coulomb repulsion. However, the strong geometrical frustration of the triangular lattice can prohibit long-range CO, resulting in a charge-glass state in which the charge configurations are randomly distributed. Here, we investigate the charge-glass states of orthorhombic and monoclinic θ-type BEDT-TTF salts by measuring the electrical resistivity and optical conductivity spectra. We find a substantial difference between the charge-glass states of the orthorhombic and monoclinic systems. The charge-glass state in the orthorhombic system with an isotropic triangular lattice exhibits larger low-energy excitations than that in the monoclinic one with an anisotropic triangular lattice and becomes more metallic as the isotropy of the triangular lattice increases. These results can be understood by the different charge-glass formation mechanisms in the two systems: in the orthorhombic system, the charge-glass state originates from geometric frustration due to the equilateral triangular lattice, leading to metallic 3-fold COs, whereas in the monoclinic system, the charge-glass formation originates from geometric frustration of the isosceles triangular lattice, in which the charge-glass state is described by the superposition of insulating 2-fold stripe COs.

Non-magnetic ion site disorder effects on the quantum magnetism of a spin-1/2 equilateral triangular lattice antiferromagnet

With the motivation to study how non-magnetic ion site disorder affects the quantum magnetism of Ba3CoSb2O9, a spin-1/2 equilateral triangular lattice antiferromagnet, we performed DC and AC susceptibility, specific heat, elastic and inelastic neutron scattering measurements on single crystalline samples of Ba2.87Sr0.13CoSb2O9 with Sr doping on non-magnetic Ba2+ ion sites. The results show that Ba2.87Sr0.13CoSb2O9 exhibits (i) a two-step magnetic transition at 2.7 K and 3.3 K, respectively; (ii) a possible canted 120 degree spin structure at zero field with reduced ordered moment as 1.24 μ B/Co; (iii) a series of spin state transitions for both H∥ab-plane and H∥c-axis. For H∥ab-plane, the magnetization plateau feature related to the up–up–down phase is significantly suppressed; (iv) an inelastic neutron scattering spectrum with only one gapped mode at zero field, which splits to one gapless and one gapped mode at 9 T. All these features are distinctly different from those observed for the parent compound Ba3CoSb2O9, which demonstrates that the non-magnetic ion site disorder (the Sr doping) plays a complex role on the magnetic properties beyond the conventionally expected randomization of the exchange interactions. We propose the additional effects including the enhancement of quantum spin fluctuations and introduction of a possible spatial anisotropy through the local structural distortions.

High-field magnetic structure of the triangular antiferromagnet RbFe(MoO4)2

The magnetic H - T phase diagram of a quasi-two-dimensional antiferromagnet RbFe(MoO4)2 with an equilateral triangular lattice structure is studied with 87Rb NMR and neutron diffraction techniques. This combination of experimental techniques allows us to determine the ordered components of the magnetic moments on the Fe3+ ions within various high-field phases - the Y, UUD, V, and fan structures, stabilized in the compound by the in-plane magnetic field. It is also established that the transition from the V to the fan phase is of first-order, whereas the transition from the fan phase to the polarized paramagnetic phase is continuous. An analysis of the NMR spectra shows that the high-field fan phase of RbFe(MoO4)2 can be successfully described by a periodic commensurate oscillation of the magnetic moments around the field direction in each Fe layer combined with an incommensurate modulation of the magnetic structure perpendicular to the layers.

Spectral MoM NUFFT-Based Formulation for the Efficient Analysis of High-Order Bandpass FSSs With Tightly Packed Nonresonant Elements in Skewed Grid

This communication presents a formulation of the spectral-domain Method of Moments (spectral MoM) for the analysis of periodic multilayered structures with a number of metallized interfaces alternating patches and apertures. Edge singularity entire-domain basis functions are used to model the electric currents on the patches and the tangential electric fields in the apertures. The formulation is especially suitable to characterize high-order bandpass frequency selective surfaces (FSSs) made up of nonresonant elements. Also in this communication, the nonuniform fast Fourier transform (NUFFT) is used in the spectral MoM to cover the case of unit cells in skewed grids, which enables the analysis of nonrectangular periodic lattices with tightly packed elements. Numerical examples of the design of third-order and fifth-order bandpass FSSs are presented. The FSSs are made up of hexagonal patches and apertures arranged in an equilateral triangular lattice. Results are cross-checked against commercial software CST Microwave Studio (CST MWS), and excellent agreement is found, our in-house software being more than one order of magnitude faster than CST MWS. This CPU time saving makes the proposed formulation very convenient for full-wave optimization and design.

Quantum spin state transitions in the spin-1 equilateral triangular lattice antiferromagnet Na2BaNi(PO4)2

We have grown single crystals of Na$_2$BaNi(PO$_4$)$_2$, a new spin-1 equilateral triangular lattice antiferromagnet (ETLAF), and performed magnetic susceptibility, specific heat and thermal conductivity measurements at ultralow temperatures. The main results are (i) at zero magnetic field, Na$_2$BaNi(PO$_4$)$_2$ exhibits a magnetic ordering at 430 mK with a weak ferromagnetic moment along the $c$ axis. This suggests a canted 120$^\circ$ spin structure, which is in a plane including the crystallographic $c$ axis due to the existence of an easy-axis anisotropy and ferromagnetically stacked along the $c$ axis; (ii) with increasing field along the $c$ axis, a 1/3 magnetization plateau is observed which means the canted 120$^\circ$ spin structure is transformed to a up up down (UUD) spin structure. With even higher fields, the UUD phase further evolves to possible V and V' phases; (iii) with increasing field along the $a$ axis, the canted 120$^\circ$ spin structure is possibly transformed to a umbrella phase and a V phase. Therefore, Na$_2$BaNi(PO$_4$)$_2$ is a rare example of spin-1 ETLAF with single crystalline form to exhibit easy-axis spin anisotropy and series of quantum spin state transitions.

Mean wind loads on equilateral triangular lattice tower under skewed wind loading

Compared with lattice towers with square or rectangle cross sections, triangular lattice towers have the advantage of smaller foundation footprint. Triangular lattice towers have been widely used in transmission towers and communication towers, yet only a few studies have been conducted on their wind loads. Therefore, a series of wind tunnel tests on a triangular lattice tower were conducted using direct force measurements in this study. Effective drag coefficients and effective skewed wind load factors normalized with the “true” windward projected areas were proposed and obtained from the test results. Piecewise functions were suggested to calculate the effective skewed wind load factors and skewed wind load factors. It is found that the skewed wind load factors show an approximate “W” profile which the British standard formula does not fit. The effective skewed wind load factors show a strong regularity and are only related to yaw angles and solidity ratios. The analytical effective skewed wind load factors have a good agreement with fitting results with a maximum error of 3.1%. The recommended formula for skewed wind load factors is concise and can effectively illustrate the changing tendency of drag coefficient curves.

Dual-Polarized Planar-Printed Ultrawideband Antenna Array on a Triangular Grid

This article presents a dual-polarized planar-printed ultrawideband antenna array on an optimally sampled triangular (hexagonal) lattice. The design is based on the dual-polarized planar ultrawideband modular antenna (PUMA). In this article, the dual-polarized square lattice PUMA array is adapted to an equilateral triangular lattice for grating-lobe-free scanning at a 15.5% reduced element count. A prototype triangular lattice PUMA array is built and compared with a square lattice PUMA aperture of similar size. As configured, the triangular lattice array demonstrates an 8% larger aperture area with 7% fewer elements than the square lattice array. Modeling predictions are validated with measured results for embedded element performance. The triangular lattice PUMA is shown to compare favorably to the square lattice PUMA with modest performance compromises.

Crystal-field Hamiltonian and anisotropy in KErSe2 and CsErSe2

We use neutron scattering and bulk property measurements to determine the single-ion crystal-field Hamiltonians of delafossites $\rm KErSe_2$ and $\rm CsErSe_2$. These two systems contains planar equilateral triangular Er lattices arranged in two stacking variants: rhombohedral (for K) or hexagonal (Cs). Our analysis shows that regardless the stacking order both compound exhibit an easy-plane ground state doublet with large $J_z=1/2$ terms and the potential for significant quantum effects, making them candidates for quantum spin liquid or other exotic ground states.

Design of low dispersion and low loss photonic crystal fiber: Defected core circular-octagon hybrid lattices

• Simulation studies are carried out on a circular-octagon hybrid PCF. • Air hole defects were introduced in the fiber core to reduce dispersion. • The proposed hybrid PCF exhibited negative dispersion with a low confinement loss. Photonic crystal fibers consisting of a novel hybrid structure providing low values for both dispersion and confinement loss were simulated using finite element method. The studies on different arrangements, polygonal and circular, of air holes as cladding of PCFs have shown that circular PCFs results in a structure with low dispersion, air filling fractions remaining same. Circular PCF with dispersion 103.50 ps/nm.km and confinement loss 5.97 × 10 −6 dB/m was restructured by replacing its first ring with octagonal air hole lattice having eight air holes. This hybrid circular-octagon PCF has shown a dispersion of 33.15 ps/nm.km and a confinement loss of 2.52 × 10 −6 dB/m. The introduction of air hole defects in the hybrid fiber core with three identical air holes in an equilateral triangular lattice was found to modify dispersion by reducing its value to −11.63 ps/nm.km with confinement loss 9.51 × 10 −5 dB/m.

Simultaneous low-loss and low-dispersion in a photonic-crystal waveguide for terahertz communications

Conventional photonic-crystal waveguides make use of an equilateral triangular lattice of through holes, here we develop an isosceles triangular lattice photonic-crystal waveguide based on a silicon slab at 0.3 terahertz (THz) band, for THz high-speed communications. The propagation loss of the proposed waveguide is as small as ∼1/10, compared with that of a conventional waveguide under the conditions of broadband bandwidth (>20 GHz) for both the loss and dispersion, due to the broadband low-dispersion below the light line of air, at which low-loss conditions are satisfied. Finally, we demonstrate 36 Gbit s−1 error-free THz communications using the isosceles triangular lattice photonic-crystal waveguide.

Crystal growth, crystal structure and anisotropic magnetic properties of KBaR(BO3)2 (R = Y, Gd, Tb, Dy, Ho, Tm, Yb and Lu) triangular lattice materials

Abstract Crystals of the KBaR(BO3)2 (R = Y, Gd, Tb, Dy, Ho, Tm, Yb and Lu) rare earth borates were grown by spontaneous nucleation from a KF-B2O3 flux. The crystals obtained were typically well-developed hexagonal plates about 1 mm in large dimension. The crystals were used to study the anisotropic temperature and field dependent magnetization of the materials, which are based on ideal triangular planes of magnetic rare earths. All structures were refined in space group R 3 - m by single-crystal X-ray diffraction, and in addition to the equilateral triangular rare earth plane lattice, displayed K/Ba site occupancy disorder. All magnetic compounds reveal dominantly antiferromagnetic spin interactions with no magnetic ordering above 1.8 K. With the exception of KBaGd(BO3)2, magnetic anisotropy was observed for other five magnetic members of the family. The results suggest that KBaR(BO3)2 crystals may be of further interest for investigating the magnetic properties of rare earth ions on a triangular lattice.

Wave propagation in non-centrosymmetric beam-lattices with lumped masses: Discrete and micropolar modeling

The in-plane acoustic behavior of non-centrosymmetric lattices having nodes endowed with mass and rotational inertia and connected by massless ligaments with asymmetric elastic properties has been analyzed through a discrete model and a continuum micropolar model. In the first case the propagation of harmonic waves and the dispersion functions have been obtained by the discrete Floquet–Bloch approach. It is shown that the optical branch departs from a critical point with vanishing group velocity and is decreasing for increasing the norm of the wave vector. A micropolar continuum model has been derived through a continualization method based on a down-scaling law from a second-order Taylor expansion of the generalized macro-displacement field. It is worth noting that the second order elasticity tensor coupling curvatures and micro-couples turns out to be negative-definite also in the general case of non-centrosymmetric lattice. The eigenvalue problem governing the harmonic propagation in the micropolar non-centrosymmetric continuum results in general characterized by a hermitian full matrix that is exact up to the second order in the wave vector.Examples concerning square and equilateral triangular lattices have been analyzed and their acoustic properties have been derived with the discrete and continuum models. The dependence of the Floquet–Bloch spectra on the lattice non-centrosymmetry is shown together with validity limits of the micropolar model. Finally, in consideration of the negative definiteness of the second order elastic tensor of the micropolar model, the loss of strong hyperbolicity of the equation of motion has been investigated.

Magnetism and multiferroicity of an isosceles triangular lattice antiferromagnet Sr3NiNb2O9

Various experimental measurements were performed to complete the phase diagram of a weakly distorted triangular lattice system, Sr3NiNb2O9 with Ni2+ , spin-1 magnetic ions. This compound possesses an isosceles triangular lattice with two shorter bonds and one longer bond. It shows a two-step magnetic phase transition at K and K at zero magnetic field, characteristic of an easy-axis anisotropy. In the magnetization curves, a series of magnetic phase transitions was observed such as an up-up-down phase at T with 1/3 of the saturation magnetization (Msat) and an oblique phase at T with /3 Msat. Intriguingly, the magnetic phase transition below TN2 is in tandem with the ferroelectricity, which demonstrates multiferroic behaviors. Moreover, the multiferroic phase persists in all magnetically ordered phases regardless of the spin structure. The comparison between the phase diagrams of Sr3NiNb2O9 and its sister compound with an equilateral triangular lattice antiferromagnet Ba3NiNb2O9 (Hwang et al 2012 Phys. Rev. Lett. 109 257205), illustrates how a small imbalance among exchange interactions change the magnetic ground states of the TLAFs.

Resonance-enhanced collective effect in a triangle arrangement of Rydberg atoms with anisotropic interactions

We investigate the collective excitation effect in a scheme where three identical Rydberg atoms are arranged in an equilateral triangular lattice. By using a static electric field polarizing the atomic dipoles, the dipole-dipole interactions between two Rydberg atoms are essentially anisotropic and can even disappear in the several special resonance cases. For that fact, we observe collectively enhanced excitation probability of single Rydberg atom in resonant areas in the case of strong blockade, and that of double or triple Rydberg atoms in the case of partial blockade. To give more evidences for this collective excitation enhancement, we study the two-body quantum correlation between three Rydberg atoms, as well as the dependence of the blockade radius on the length of triangle sides, which present a good agreement with the excitation properties.

Photonic crystal-based optical filter: a brief investigation

Ring resonator-based channel drop filters are investigated conceptually and analytically with different ring (square and hexagonal) structures. In the proposed architecture, silicon rods ($$n=3.4641$$n=3.4641) are contrived over an air substrate of refractive index $$n=1$$n=1 in the equilateral triangular lattice which has the lattice constant $$a=900\,\hbox {nm}$$a=900nm. The characteristics of the design are examined for various types of pillars (circular and elliptical) and are reported in the analysis. The band gap for each structure is calculated and observed by plane-wave expansion method. The normalized transmission spectra and resonance wavelengths for different photonic crystal ring resonators are obtained using 2D finite-difference time-domain method. From the investigation, the resonance of circular pillar falls over the region of the third window at C-band (1530---1565 nm) which has the lowest attenuation losses and most widely used. Full width at half maximum and quality factor of the desired layout are also being calculated. The size of the device is about $$20 \times 14\,{\upmu }\hbox {m}$$20×14μm which is highly compact and useful for the integration of photonic circuits.

Static and Dynamical Properties of the Spin-1/2 Equilateral Triangular-Lattice Antiferromagnet Ba_{3}CoSb_{2}O_{9}.

We present single-crystal neutron scattering measurements of the spin-1/2 equilateral triangular-lattice antiferromagnet Ba_{3}CoSb_{2}O_{9}. Besides confirming that the Co^{2+} magnetic moments lie in the ab plane for zero magnetic field and then determining all the exchange parameters of the minimal quasi-2D spin Hamiltonian, we provide conclusive experimental evidence of magnon decay through observation of intrinsic line broadening. Through detailed comparisons with the linear and nonlinear spin-wave theories, we also point out that the large-S approximation, which is conventionally employed to predict magnon decay in noncollinear magnets, is inadequate to explain our experimental observation. Thus, our results call for a new theoretical framework for describing excitation spectra in low-dimensional frustrated magnets under strong quantum effects.

Ising antiferromagnet on a finite triangular lattice with free boundary conditions

The exact integer values for the density of states of the Ising model on an equilateral triangular lattice with free boundary conditions are evaluated up to L = 24 spins on a side for the first time by using the microcanonical transfer matrix. The total number of states is 2Ns = 2300 ≈ 2.037 × 1090 for L = 24, where Ns = L(L+1)/2 is the number of spins. Classifying all 2300 spin states according to their energy values is an enormous work. From the density of states, the exact partition function zeros in the complex temperature plane of the triangular-lattice Ising model are evaluated. Using the density of states and the partition function zeros, we investigate the properties of the triangularlattice Ising antiferromagnet. The scaling behavior of the ground-state entropy and the form of the correlation length at T = 0 are studied for the triangular-lattice Ising antiferromagnet with free boundary conditions. Also, the scaling behavior of the Fisher edge singularity is investigated.

General selection of lattice shape in ductile-reinforced brittle structures for increased stiffness

Ductile reinforcements are used in brittle structures to increase the overall stiffness and strength. However, the optimum lattice shape geometry according to which the ductile reinforcements must be laid down, has not been generally established. In this work, a ductile isotropic lattice structure is constructed from a unit cell and the brittle matrix is “filled” in the empty spaces thereafter. In homogeneous stress conditions, the brittle part will experience the same applied remote stress as is felt by the ductile lattice. The statistical theory of brittle failure is invoked to bring home the point that the unit cell area of the lattice is arbitrary (for a given total area) and hence the lattice shape geometry does not influence the probability of survival of the structure. Therefore, a suitable geometry shape (triangle or square) maybe chosen to further enhance the overall stiffness at a given probability of survival. It has been concluded that an (equilateral) triangular lattice is better suited compared to a square lattice if the ductile reinforcement lattice bars have the same stiffness. The technique and conclusion is generally true for any choice of brittle and ductile materials. Also, the conclusion holds true for both compressive and tensile loading.