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In this research, Al-doped (with 2, 4, 6, and 8 mol%) zinc oxide thin films deposited onto glass substrates using wet chemical spin coating technique were characterized via X-ray diffraction (XRD), scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS), and UV–Vis spectroscopic methods to realize the doping effect of Al on structural and optical properties of ZnO thin films. XRD analysis revealed that the Al-doped ZnO (AZO) thin films were polycrystalline in nature with hexagonal lattice structure, and crystallite growth along c -axis. SEM images on the film surface at lower Al content exhibited granular nanostructures. At higher Al content, the nanostructural features of ZnO thin films was observed to be diminished. The compositional analysis via EDS measurements indicated the presence of zinc, oxygen, and aluminium into the AZO thin films. AZO thin film containing 2 mol% Al exhibited higher transmittance of 97% and absorption edge at 385 nm in the visible region. Optical bandgaps of AZO thin films were varied within 3.30–3.37 eV. Noticeable variations on structural features, and optical parameters of AZO thin films were detected, and the changing trend showed was non-linear and non-monotonic in nature. Graphic Abstract

We consider planar elastic beam Hamiltonians defined on hexagonal lattices. These quantum graphs are constructed from Euler–Bernoulli beams, each governed by the fourth-order Schrödinger operator with a real periodic symmetric potential function. In contrast to the second-order Schrödinger operator commonly studied in the quantum graph literature, here vertex matching conditions encode the geometry of the underlying graph by their dependence on angles at which the edges meet. We show that on the hexagonal lattice, the dispersion relation has a structure similar to that reported for the periodic second-order Schrödinger operator, known as the “graphene Hamiltonian.” This property is then utilized to prove the existence of Dirac points (conical singularities). We further discuss the (ir)reducibility of Fermi surfaces. Moreover, we obtain the point spectrum, the absolutely continuous spectrum, and the singular continuous spectrum. Applying perturbation analysis, we derive the dispersion relation for the planar elastic beam Hamiltonians on angle-perturbed irregular hexagonal lattices, defined in a geometric neighborhood of the hexagonal lattice. On these graphs, we find that, unlike the hexagonal lattice, the dispersion relation is not split into purely energy- and quasimomentum-dependent terms; however, Dirac points exist similar to the hexagonal-lattice case.

AbstractPhase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice are investigated by the Mote Carlo method based on the Wang–Landau algorithm. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

Sr0.5Ni0.5LaxFe12 − xO19 (x = 0, 0.08, 0.16, and 0.24) hexaferrites have been synthesized by ball-milling-assisted ceramic process. The calcined samples are characterized by X-ray powder diffraction (XRD), scanning electron microscopy (SEM), Fourier transform infrared spectra (FTIR), and vibrating sample magnetometry (VSM). XRD reveals that all samples consist of the main M-type hexagonal Sr ferrite phase and a low amount of rhombohedral Fe2O3 as secondary phase, and a continuous increase of the hexagonal lattice parameters with substitution content (x). Magnetic characterization indicates that all Sr0.5Ni0.5LaxFe12 − xO19 samples have a hard magnetic property. Substitution of Fe3+ ions by La3+ ions can obviously improve the specific saturation magnetization, remanence, magnetic moment, and effective anisotropy constant. Specific saturation magnetization, remanence, and the magnetic moment reach a maximum at x= 0.24 and calcination temperature of 1050 ∘C.

Crystal lattices with tetragonal or hexagonal structure often exhibit structural transitions in response to external stimuli 1 . Similar behaviour is anticipated for the lattice forms of topological spin textures, such as lattices composed of merons and antimerons or skyrmions and antiskyrmions (types of vortex related to the distribution of electron spins in a magnetic field), but has yet to be verified experimentally 2 , 3 . Here we report real-space observations of spin textures in a thin plate of the chiral-lattice magnet Co 8 Zn 9 Mn 3 , which exhibits in-plane magnetic anisotropy. The observations demonstrate the emergence of a two-dimensional square lattice of merons and antimerons from a helical state, and its transformation into a hexagonal lattice of skyrmions in the presence of a magnetic field at room temperature. Sequential observations with decreasing temperature reveal that the topologically protected skyrmions remain robust to changes in temperature, whereas the square lattice of merons and antimerons relaxes to non-topological in-plane spin helices, highlighting the different topological stabilities of merons, antimerons and skyrmions. Our results demonstrate the rich variety of topological spin textures and their lattice forms, and should stimulate further investigation of emergent electromagnetic properties. A magnetically induced two-dimensional square lattice of merons and antimerons is observed in real space, along with its transformation into a hexagonal lattice of skyrmions at room temperature.

Two-dimensional materials with a hexagonal lattice, such as graphene, have two distinct “valleys” in their band structure. Researchers in the emerging field of valleytronics hope that these valley degrees of freedom can be exploited as information carriers, but making valleytronic devices is tricky. Li et al. created chiral valley Hall states on the boundary between oppositely gated regions of bilayer graphene. They then guided these so-called kink states through their sample using spatially modulated gating, demonstrating right and left turns, as well as a valley valve function. Science , this issue p. 1149 Spatially modulated gating of a bilayer graphene sample controls the propagation of chiral valley Hall currents. Developing alternative paradigms of electronics beyond silicon technology requires the exploration of fundamentally new physical mechanisms, such as the valley-specific phenomena in hexagonal two-dimensional materials. We realize ballistic valley Hall kink states in bilayer graphene and demonstrate gate-controlled current transmission in a four-kink router device. The operations of a waveguide, a valve, and a tunable electron beam splitter are demonstrated. The valley valve exploits the valley-momentum locking of the kink states and reaches an on/off ratio of 8 at zero magnetic field. A magnetic field enables a full-range tunable coherent beam splitter. These results pave a path to building a scalable, coherent quantum transportation network based on the kink states.

Topological phases constitute an exotic form of matter characterized by non-local properties rather than local order parameters1. The paradigmatic Haldane model on a hexagonal lattice features such topological phases distinguished by an integer topological invariant known as the first Chern number2. Recently, the identification of non-equilibrium signatures of topology in the dynamics of such systems has attracted particular attention3–6. Here, we experimentally study the dynamical evolution of the wavefunction using time- and momentum-resolved full state tomography for spin-polarized fermionic atoms in driven optical lattices7. We observe the appearance, movement and annihilation of dynamical vortices in momentum space after sudden quenches close to the topological phase transition. These dynamical vortices can be interpreted as dynamical Fisher zeros of the Loschmidt amplitude8, which signal a so-called dynamical phase transition9,10. Our results pave the way to a deeper understanding of the connection between topological phases and non-equilibrium dynamics.

Background Studies using quantitative experimental methods have shown that intracellular spatial distribution of molecules plays a central role in many cellular systems. Spatially resolved computer simulations can integrate quantitative data from these experiments to construct physically accurate models of the systems. Although computationally expensive, microscopic resolution reaction-diffusion simulators, such as Spatiocyte can directly capture intracellular effects comprising diffusion-limited reactions and volume exclusion from crowded molecules by explicitly representing individual diffusing molecules in space. To alleviate the steep computational cost typically associated with the simulation of large or crowded intracellular compartments, we present a parallelized Spatiocyte method called pSpatiocyte. Results The new high-performance method employs unique parallelization schemes on hexagonal close-packed (HCP) lattice to efficiently exploit the resources of common workstations and large distributed memory parallel computers. We introduce a coordinate system for fast accesses to HCP lattice voxels, a parallelized event scheduler, a parallelized Gillespie’s direct-method for unimolecular reactions, and a parallelized event for diffusion and bimolecular reaction processes. We verified the correctness of pSpatiocyte reaction and diffusion processes by comparison to theory. To evaluate the performance of pSpatiocyte, we performed a series of parallelized diffusion runs on the RIKEN K computer. In the case of fine lattice discretization with low voxel occupancy, pSpatiocyte exhibited 74% parallel efficiency and achieved a speedup of 7686 times with 663552 cores compared to the runtime with 64 cores. In the weak scaling performance, pSpatiocyte obtained efficiencies of at least 60% with up to 663552 cores. When executing the Michaelis-Menten benchmark model on an eight-core workstation, pSpatiocyte required 45- and 55-fold shorter runtimes than Smoldyn and the parallel version of ReaDDy, respectively. As a high-performance application example, we study the dual phosphorylation-dephosphorylation cycle of the MAPK system, a typical reaction network motif in cell signaling pathways. Conclusions pSpatiocyte demonstrates good accuracies, fast runtimes and a significant performance advantage over well-known microscopic particle methods in large-scale simulations of intracellular reaction-diffusion systems. The source code of pSpatiocyte is available at https://spatiocyte.org .

The sp 2 nature of graphene endows the hexagonal lattice with very high theoretical stiffness, strength and resilience, all well-documented. However, the ultimate stretchability of graphene has not yet been demonstrated due to the difficulties in experimental design. Here, directly performing in situ tensile tests in a scanning electron microscope after developing a protocol for sample transfer, shaping and straining, we report the elastic properties and stretchability of free-standing single-crystalline monolayer graphene grown by chemical vapor deposition. The measured Young’s modulus is close to 1 TPa, aligning well with the theoretical value, while the representative engineering tensile strength reaches ~50-60 GPa with sample-wide elastic strain up to ~6%. Our findings demonstrate that single-crystalline monolayer graphene can indeed display near ideal mechanical performance, even in a large area with edge defects, as well as resilience and mechanical robustness that allows for flexible electronics and mechatronics applications. The extraordinary mechanical properties of graphene are usually measured on very small or supported samples. Here, the authors develop a method to test a large area of graphene and show that even with edge defects it displays near-ideal mechanical performance.

We investigate elastic periodic structures characterized by topologically nontrivial bandgaps supporting backscattering suppressed edge waves. These edge waves are topologically protected and are obtained by breaking inversion symmetry within the unit cell. Examples for discrete one and two-dimensional lattices elucidate the concept and illustrate parallels with the quantum valley Hall effect. The concept is implemented on an elastic plate featuring an array of resonators arranged according to a hexagonal topology. The resulting continuous structures have non-trivial bandgaps supporting edge waves at the interface between two media with different topological invariants. The topological properties of the considered configurations are predicted by unit cell and finite strip dispersion analyses. Numerical simulations demonstrate edge wave propagation for excitation at frequencies belonging to the bulk bandgaps. The considered plate configurations define a framework for the implementation of topological concepts on continuous elastic structures of potential engineering relevance.