Two-dimensional Order Research Articles

Spin and lattice dynamics of the two-dimensional van der Waals ferromagnet CrI3

Chromium tri-iodide (CrI3) is a prototypical ferromagnetic van der Waals insulator with its genuinely two-dimensional (2D) long-range magnetic order below 45 K demonstrated recently. The underlying magnetic anisotropy has not been completely understood while both the Dzyaloshinskii-Moriya (DM) interaction and the Kitaev−Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Gamma$$\\end{document} type interaction have been considered as the relevant magnetic Hamiltonian. In addition, the relation between the crystal structure and the magnetic order needs to be further elucidated concerning their possible coupling in few-layer samples and in the topmost surface layers of bulk samples. Here, we investigate these issues via temperature- and magnetic field-dependent terahertz spectroscopy on bulk CrI3 single crystals, focusing on the dynamics of ferromagnetic resonances (FMRs) and optical phonons in the terahertz (THz) region from 4 to 120 cm−1 (from 0.5 to 15 meV). We narrow down the possible ranges of the interaction parameters such as the off-diagonal symmetric terms and the single-ion anisotropy. The accurate values of these parameters significantly constrain the magnitude of possible Kitaev−Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Gamma$$\\end{document} exchange interaction and the topological magnon gap. Moreover, the structure-magnetism relationship was critically analyzed based on the temperature- and field-dependences of two Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\rm{E}}}_{{\\rm{u}}}$$\\end{document} in-plane optical phonon modes, which shows that the commonly believed structural phase diagram of CrI3, derived from surface-preferential data, has to be seriously modified.

Coexistence of Ferromagnetism and Superconductivity at KTaO3 Heterointerfaces.

The coexistence of superconductivity and ferromagnetism is a long-standing issue in superconductivity due to the antagonistic nature of these two ordered states. Experimentally identifying and characterizing novel heterointerface superconductors that coexist with magnetism presents significant challenges. Here, we report the observation of two-dimensional long-range ferromagnetic order in a KTaO3 heterointerface superconductor, showing the coexistence of superconductivity and ferromagnetism. Remarkably, our direct current superconducting quantum interference device measurements reveal an in-plane magnetization hysteresis loop persisting above room temperature. Moreover, first-principles calculations and X-ray magnetic circular dichroism measurements provide decisive insights into the origin of the observed robust ferromagnetism, attributing it to oxygen vacancies that localize electrons in nearby Ta 5d states. Our findings suggest KTaO3 heterointerfaces as time-reversal symmetry breaking superconductors, injecting fresh momentum into the exploration of the intricate interplay between superconductivity and magnetism enhanced by the strong spin-orbit coupling inherent to the heavy Ta in 5d orbitals.

Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking

Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking

Highly Ordered Monolayers of μm-Sized Polystyrene Spheres Studied by Grazing-Incidence Small-Angle X-ray Scattering, Simulations, and Geometrical Calculations.

Unraveling the two-dimensional (2D) structural ordering of colloidal particles assembled at a flat surface is essential for understanding and optimizing their physical properties. So far, grazing-incidence small-angle X-ray scattering (GISAXS) has been widely used to determine crystallographic information on 2D self-assembled structures of nanosize objects. However, solving the structure of 2D lattices consisting of micrometer (μm)-sized objects still remains a challenge using scattering methods. Here, a model 2D SCALMS (supported catalytically active liquid metal solution) template is fabricated from μm-sized polystyrene (PS) spheres that form a monolayer on top of the flat solid support. GISAXS patterns of the sample were collected for rotation angles around its surface normal in steps of 3°. For every rotation angle, different Bragg-type interference maxima along the out-of-plane (qz) direction were observed. On the basis of simulations of GISXAS patterns of single domains of ordered particle arrangements using the distorted wave Born approximation (DWBA) and validation against a simple geometrical scattering model, the interference maxima could nicely be interpreted to originate from a monolayer of the μm-sized spherical particles which are arranged in domains of hexagonal 2D paracrystalline order. This novel GISAXS evaluation technique serves as a proof of principle for determining the μm-size periodicity of 2D crystalline domains and demonstrates its potential to spatially resolve the relative orientations of such domains with respect to a reference direction.

Controlling the 2D Magnetism of CrBr3 by van der Waals Stacking Engineering.

The manipulation of two-dimensional (2D) magnetic order is of significant importance to facilitate future 2D magnets for low-power and high-speed spintronic devices. van der Waals stacking engineering makes promises for controllable magnetism via interlayer magnetic coupling. However, directly examining the stacking order changes accompanying magnetic order transitions at the atomic scale and preparing device-ready 2D magnets with controllable magnetic orders remain elusive. Here, we demonstrate the effective control of interlayer stacking in exfoliated CrBr3 via thermally assisted strain engineering. The stable interlayer ferromagnetic (FM), antiferromagnetic (AFM), and FM-AFM coexistent ground states confirmed by the magnetic circular dichroism measurements are realized. Combined with the first-principles calculations, the atomically resolved imaging technique reveals the correlation between magnetic order and interlayer stacking order in CrBr3 flakes unambiguously. A tunable exchange bias effect is obtained in the mixed phase of FM and AFM states. This work will introduce new magnetic properties by controlling the stacking order and sequence of 2D magnets, providing ample opportunities for their application in spintronic devices.

Two-dimensional ordering of bacteriorhodopsins in a lipid bilayer and effects caused by repulsive core between lipid molecules on lateral depletion interaction: A study based on a thermodynamic perturbation theory

Using binary hard disk mixture models, we studied the two-dimensional ordering of bacteriorhodopsins in a lipid bilayer. The phase diagrams were calculated using the thermodynamic perturbation theory. We examined two types of effective interactions to discuss the lateral depletion effects caused by repulsive core interaction between lipid molecules. The results indicate that the core repulsions drastically broaden the coexistence region for the fluid-ordered phase.

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

Radial Alignment of Carbon Nanotubes Via Dead-End Filtration

Dead-end filtration is a facile method to globally align single wall carbon nanotubes (SWCNTs) in large area films with a two-dimensional order parameter, , approaching unity. Uniaxial alignment has been achieved using pristine and hot-embossed membranes but more sophisticated geometries have yet to be investigated. In this work, we create three different patterns with radial symmetry and an area of 3.8 cm². Two of these patterns are replicated by the filtered SWCNTs and values of » 0.85 are obtained. Each of the radially aligned SWCNT films are characterized by scanning cross-polarized microscopy in reflectance and laser imaging in transmittance with linear, radial and azimuthal polarized light fields. The former is used to define a novel indicator akin to the two-dimensional order parameter using Malu’s law, yielding 0.82 for the respective film. The films are then transferred to a flexible printed circuit board and terminal two-probe electrical measurements are conducted to explore the potential of those new alignment geometries. Figure 1

(Invited) The Impact of Carbon Nanotube Length and Diameter on Their Global Alignment by Dead-End Filtration

Dead-end filtration on polyvinylpyrrolidone-coated polycarbonate track-etched membranes has proven to be an effective method to prepare macroscopically (3.8 cm²) aligned thin films from solution-based single-wall carbon nanotubes (SWCNTs). However, to make this technique broadly applicable, the role of SWCNT length and diameter on the alignment process must be understood. To date, most groups report the alignment of unsorted large diameter (» 1.4 nm) SWCNTs, but systematic studies on their small diameter (» 0.78 nm) counterparts are rare and in the case of sorted SWCNT species, non-existent. In this work, films are prepared from two series of length sorted fractions comprised of small and large diameter SWCNTs, respectively. The degree of alignment is characterized by cross polarized microscopy, scanning electron microscopy and Raman spectroscopy. For the longest fractions, the two-dimensional order parameter, , values of » 0.66 and » 0.76 are reported for the small and large diameter SWCNTs, respectively, and Raman maps over an area of 2500 μm2 are presented. A comparison of Derjaguin, Landau, Verwey and Overbeek (DLVO) theory calculations with the aligned domain size is then used to propose a law identifying the required length of a carbon nanotube with a given diameter and zeta potential for alignment.

Two-Dimensional Fractional Order Iterative Learning Control for Repetitive Processes

The convergence and robustness rejecting parameters variations and external disturbance of the system are crucial for repetitive processes. In this paper, a two-dimensional robust fractional-order iterative learning control (FOILC) is proposed for the repetitive motion process to enhance the convergence and robustness. A fractional-order proportional derivative function (FOPDF) is designed as the control variable to replace the tracking error of the integer-order iterative learning control (IOILC) algorithm. The required dynamic output fractional-order iterative learning controller is constructed by solving a set of linear matrix inequalities (LMI), and the control parameters are adjusted according to the given specifications. Simulation and experimental results in robot torque control are given to prove the effectiveness and feasibility of the proposed design method.

Dynamical theory of topological defects I: the multivalued solution of the diffusion equation

Point-like topological defects are singular configurations that manifest in and out of various equilibrium systems with two-dimensional orientational order. Because they are associated with a nonzero circuitation condition, the presence of defects induces a long-range perturbation of the orientation landscape around them. The effective dynamics of defects is thus generally described in terms of quasi-particles interacting via the orientation field they produce, whose evolution in the simplest setting is governed by the diffusion equation. Because of the multivalued nature of the orientation field, its expression for a defect moving with an arbitrary trajectory cannot be determined straightforwardly and is often evaluated in the quasi-static approximation. Here, we instead derive the exact expression for the orientation created by multiple moving defects, which we find to depend on their past trajectories and thus to be nonlocal in time. Performing various expansions in relevant regimes, we demonstrate how improved approximations with respect to the quasi-static defect solution can be obtained. Moreover, our results lead to so far unnoticed structures in the orientation field of moving defects, which we discuss in light of existing experimental results.

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice.

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

Spin-phonon coupling in two-dimensional magnetic materials

Recently, two-dimensional magnetic materials (2DMMs) have become a focused research direction in a broad range of two-dimensional materials, due to their underlying significance in fundamental research, as well as in technologically relevant applications for future spintronics, magnonics, quantum information and data storage. The rich toolbox of 2DMMs and their diverse tunability have enabled the unprecedented research concerning the two-dimensional magnetic order down to single atomic layer of materials, much beyond the classical thin film magnetism, showing an extremely promising venue towards electronics, magneto-optics and photonics. Among various degrees of freedoms, the interaction between spin and phonon (i.e., quanta of lattice vibration), hence the so-called spin-phonon coupling, serves as an important tuning knob to explore the two-dimensional magnetism, create new types of quasiparticles and control the magnetic order. This review article overviews the latestdevelopments in spin-phonon coupling investigations in 2DMMs. Various techniquesutilizing spin-phononcoupling to investigate two-dimensional magnetism are discussed.Recent progresses in tuning two-dimensional magnetic order based on spin phonon-coupling are also summarized, with a focus to understand the new functionalities. Furthermore, device developments and concepts based on spin-phonon coupling are briefly discussed. This review will provide our perspectives on the existing challenges and future directions in spin-phonon coupling research in 2DMMs and their functional devices.

A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.

Collective bending motion of a two-dimensionally correlated bowl-stacked columnar liquid crystalline assembly under a shear force.

Stacked teacups inspired the idea that columnar assemblies of stacked bowl-shaped molecules may exhibit a unique dynamic behavior, unlike usual assemblies of planar disc- and rod-shaped molecules. On the basis of the molecular design concept for creating higher-order discotic liquid crystals, found in our group, we synthesized a sumanene derivative with octyloxycarbonyl side chains. This molecule forms an ordered hexagonal columnar mesophase, but unexpectedly, the columnar assembly is very soft, similar to sugar syrup. It displays, upon application of a shear force on solid substrates, a flexible bending motion with continuous angle variations of bowl-stacked columns while preserving the two-dimensional hexagonal order. In general, alignment control of higher-order liquid crystals is difficult to achieve due to their high viscosity. The present system that brings together higher structural order and mechanical softness will spark interest in bowl-shaped molecules as a component for developing higher-order liquid crystals with unique mechanical and stimuli-responsive properties.

ALTERNATIVE DIRECTION IMPLICIT METHOD FOR SOLVING FIRST ORDER 2D HYPERBOLIC DELAY DIFFERENTIAL EQUATIONS

This article considers delayed two-dimensional first order hyperbolic differential equations. The propagation of the discontinuity of the solution is also established. An alternating implicit finite difference method and backward Euler finite difference methods are presented. We proved that this method is first-order convergent. Illustrating numerical examples are given for validation. We also present an application of the proposed approach to the variable delay problem.

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann–Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss–Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method.

Recipe for higher order topology on the triangular lattice

We present a recipe for an electronic two-dimensional (2D) higher order topological insulator (HOTI) on a triangular lattice that can be realized in a large family of materials. The essential ingredient is mirror symmetry breaking, which allows for a finite quadrupole moment and trivial ${\mathbb{Z}}_{2}$ index. The competition between spin-orbit coupling and the symmetry-breaking terms gives rise to four topologically distinct phases; the HOTI phase appears when symmetry breaking dominates, including in the absence of spin-orbit coupling. We identify triangular monolayer adsorbate systems on the (111) surface of zincblende/diamond type substrates as ideal material platforms and predict the HOTI phase for $X=(\mathrm{Al},\mathrm{B},\mathrm{Ga})$ on SiC.

Two-dimensional long-range uniaxial order in three-dimensional active fluids

Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. Here, in contrast, I show that bulk active fluids have stable active nematic and polar states at fluid–fluid or fluid–air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk lead to long-range order at the interface. Thus, active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and chaotic bulk. I further consider active units that are constrained to live at an interface to examine the minimal conditions for the existence of two-dimensional order in bulk three-dimensional fluids. In this case, immotile units do not order, but motile particles still form a long-range-ordered polar phase. This prediction of stable, uniaxial, active phases in bulk fluids may have functional consequences for active transport. Bulk active fluids are unstable because activity destroys long-range ordering. Now, a model of 3D active liquids shows that stable states can form at fluid–fluid surfaces.