Six-fold Rotational Symmetry Research Articles

Oriented Organic Nanowires Self-Assembled on a Graphene Surface

Molecular self-assembly of organic molecules on two-dimensional atomic crystals involves molecule–molecule interactions, molecule–surface interactions, or their combination. Most previous studies are focused on self-assembled organic monolayers. Here, for the first time, we report oriented organic nanowires self-assembled on a graphene surface in the atmosphere. Oriented lauroyl peroxide nanowires were formed when cooling down the melting lauroyl peroxide layer on a graphene surface but were not formed on SiO2 and copper surfaces. Structural analysis and theoretical simulation revealed that the combination of strong molecule–molecule packing interactions, weak molecule–molecule linking interactions, and weak molecule–surface interactions contributed to the formation of nanowires, while the orientation of nanowires depended on the six-fold rotational symmetry of graphene and strong molecule–surface interactions.

Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials.

Metamaterials have recently enabled coupling induced transparency due to interference effects in coupled subwavelength resonators. In this work, we present a three dimensional (3-D) metamaterial design with six-fold rotational symmetry that shows electromagnetically induced transparency with a strong polarization dependence to the incident electromagnetic wave due to the ultra-sharp resonance line width as a result of interaction between the constituent meta-atoms. However, when the six-fold rotationally symmetric unit cell design was re-arranged into a fourfold rotational symmetry, we observed the excitation of a polarization insensitive dual-band transparency. Thus, the 3-D split-ring resonators allow new schemes to observe single and multi-band classical analogues of electromagnetically induced transparencies that has huge potential applications in slowing down light, sensing modalities, and filtering functionalities either in the passive mode or the active mode where such effects could be tuned by integrating materials with dynamic properties.

Design of the Planar Six-Port Crossover With Double Rings

The six-port crossover with the planar structure has been proposed. By utilizing double rings with quarter-wavelength branches, a wide passband can be obtained. With the proposed crossover, three intersecting channels with good performances of transmission and isolation can be generated. The even-odd mode and the six-fold rotational symmetry are adopted to derive the analytical equations. Moreover, the performances of diagonal transmission, port matching, and isolation between different channels are achieved with a fractional bandwidth of 13.5%, which has been demonstrated by the theoretical and measured results.

Signatures of spin-triplet excitations in optical conductivity of valence bond solids

We show that the optical responses below the Mott gap can be used to probe the spin-triplet excitations in valence bond solid (VBS) phases in Mott insulators. The optical conductivity in this regime arises due to the electronic polarization mechanism via virtual electron-hopping processes. We apply this mechanism to the Hubbard model with spin–orbit couplings and/or the corresponding spin model with significant Dzyaloshinskii–Moriya (DM) interactions, and compute the optical conductivity of VBS states on both ideal and deformed Kagome lattices. In the case of the deformed Kagome lattice, we study the antiferromagnet Rb2Cu3SnF12 with the pinwheel VBS state. In case of the ideal Kagome lattice, we explore the optical conductivity signatures of the spin-triplet excitations for three VBS states with (1) a 12-site unit cell, (2) a 36-site unit cell with six-fold rotation symmetry, and (3) a 36-site unit cell with three-fold rotation symmetry, respectively. We find that increasing the DM interactions generally leads to broad and smooth features in the optical conductivity with interesting experimental consequences. The optical conductivity reflects the features of the spin-triplet excitations that can be measured in future experiments.

Area-preserving projections from hexagonal and triangular domains to the sphere and applications to electron back-scatter diffraction pattern simulations

We construct two new area-preserving projections, which map regular hexagons and regular triangles onto circles. Combination of these projections with the inverse Lambert equal-area projection from the disc to the two hemispheres of a sphere provide bi-directional conversions between uniform planar grids with three-fold and six-fold rotational symmetry and corresponding uniform grids on the sphere. An application example is given for the representation of the channeling-modified back-scattered electron yield for hexagonal titanium.

Electronic scattering of pseudo-magnetic field induced by local bump in graphene

We investigated the electronic scattering properties of a local bump strain in graphene sheet in frame of Born approximation. The differential scattering cross section is a function of outgoing and incident angles and has the six-fold rotational symmetry with respect to both angles. The incident plane wave is scattered into two backward fan-waves in different directions in low energy limit and is split into two branches spanning the angle reversely proportional to the incident wavevector k in high energy limit. The total scattering cross section depends on incident wavevector by the form k5 in the former limit, while it is independent of k and sensitive to the incident orientation in the latter limit. We explained these features using the symmetry of the strain-induced pseudo-magnetic field.

Pattern formation in the dipolar Ising model on a two-dimensional honeycomb lattice

We present Monte Carlo simulation results for a two-dimensional Ising model with ferromagnetic nearest-neighbor couplings and a competing long-range dipolar interaction on a honeycomb lattice. Both structural and thermodynamic properties are very similar to the case of a square lattice, with the exception that structures reflect the sixfold rotational symmetry of the underlying honeycomb lattice. To deal with the long-range nature of the dipolar interaction we also present a simple method of evaluating effective interaction coefficients, which can be regarded as a more straightforward alternative to the prevalent Ewald summation techniques.

Preferential growth orientations of CuCrO2 films grown by pulsed laser deposition

We reported the investigation of the preferential growth orientations of CuCrO2 films on different substrates of (0001) Al2O3, (100) YSZ, and (100) SrTiO3 grown by pulsed laser deposition. The epitaxial thin films were grown at 700 °C with the oxygen pressure of 10 mTorr. The c-axis preferred orientation of the CuCrO2 films was observed on (000l) Al2O3 and (100) YSZ substrate. However, the CuCrO2 films on the (100) SrTiO3 substrates were oriented perpendicular to (01–15). The six-fold and twelve-fold rotational symmetry in pole figure from (01–12) of CuCrO2 films were appeared at χ = 73° using (0001) Al2O3 and (100) YSZ, respectively. The four-fold rotational symmetry points at χ = 53° were observed from the (0006) plane of CuCrO2 on (100) SrTiO3 substrate. The preferential growth orientation and epitaxial relationship of CuCrO2 films on the different substrates were explained by the lattice mismatch, surface energy, and the surface arrangement of the cation and anion.

Dependence of extinction cross-section on incident polarization state and particle orientation

This note reports on the effects of the polarization state of an incident quasi-monochromatic parallel beam of radiation and the orientation of a hexagonal ice particle with respect to the incident direction on the extinction process. When the incident beam is aligned with the six-fold rotational symmetry axis, the extinction is independent of the polarization state of the incident light. For other orientations, the extinction cross-section for linearly polarized light can be either larger or smaller than its counterpart for an unpolarized incident beam. Therefore, the attenuation of a quasi-monochromatic radiation beam by an ice cloud depends on the polarization state of the beam if ice crystals within the cloud are not randomly oriented. Furthermore, a case study of the extinction of light by a quartz particle is also presented to illustrate the dependence of the extinction cross-section on the polarization state of the incident light.

Structural properties of the epitaxial CuCr 0.95Mg 0.05O 2 thin films on c-plane sapphire substrates by pulsed laser deposition

We report an investigation of the structural properties of CuCr 0.95Mg 0.05O 2 films on c-plane sapphire substrates using pulsed laser deposition. The thin films were grown at different temperatures of 500, 600, and 700 °C with an oxygen partial pressure of 10 mTorr. c-axis oriented epitaxial CuCr 0.95Mg 0.05O 2 thin films on c-plane sapphire substrates with an in-plan 30° rotation were obtained. The sixfold rotational symmetry in the pole figures from the (0 1 2) plane indicates that there are two different types of crystal grains in which the a-axes rotate by 60° with respect to each other around the c-axis. The reason for the 30° rotation is assumed to be the presence of the ∼10% mismatch of oxygen distance between the c-plane sapphire substrate and the CuCr 0.95Mg 0.05O 2 on the c-plane. The epitaxial crystallographic relationship between CuCr 0.95Mg 0.05O 2 and Al 2O 3 was (0 0 0 6)CuCrO 2//(0 0 0 3)Al 2O 3 and [1 0 −1 0]CuCrO 2//[1 1 −2 0]Al 2O 3. The presence of twins in the films and the surface morphology were investigated using transmission electron microscopy and scanning electron microscopy, respectively.

Twin-domain Epitaxial Growth and Metal-insulator Transition of VO2 Thin Film on C-Plane Sapphire

ABSTRACTWe report heteroepitaxial growth of VO2 thin film on c-plane sapphire by pulsed DC magnetron sputtering. X-ray diffraction experiment indicates that the 150 nm thick film is in triple-domain (020)-epitaxial structure with six-fold rotational symmetry in the basal plane; in particular, off-axis Φ scans from (011) and (220) show twin and triple peaks in each group of the diffraction profiles due to angle β mismatch and V4+-V4+ dimerization, respectively. The epitaxial relationship between VO2 and c-plane sapphire can be concluded as be ${\rm{VO}}_{\rm{2}} \left[ {{\rm{010}}} \right]{\rm{ }}\left\| {{\rm{Al}}_{\rm{2}} {\rm{O}}_{\rm{3}} \left[ {{\rm{0001}}} \right]\,{\rm{and}}\,{\rm{VO}}_{\rm{2}} {\rm{ }}\left({{\rm{\bar 202}}} \right){\rm{ }}} \right\|{\rm{Al}}_{\rm{2}} {\rm{O}}_{\rm{3}} \left\{ {11\bar 20} \right\}$, with the in-plane lattice mismatch of 2.66% (tensile) along $\left[ {\bar 202} \right]$ and the out-of-plane lattice mismatch of -2.19% (compressive). Temperature dependence of resistivity in van der Pauw method shows that the resistivity changes by ~5 orders of magnitude through the metal-insulator transition, and a narrow hysteresis window of ~3 K is obtained between cooling and heating cycles with respect to phase-transition temperatures at 347.1 and 350.1 K.

Evidence for grid cells in a human memory network

Grid cells in the entorhinal cortex of freely moving rats provide a strikingly periodic representation of self-location which is indicative of very specific computational mechanisms. However, the existence of grid cells in humans and their distribution throughout the brain are unknown. Here we show that the preferred firing directions of directionally modulated grid cells in rat entorhinal cortex are aligned with the grids, and that the spatial organization of grid-cell firing is more strongly apparent at faster than slower running speeds. Because the grids are also aligned with each other, we predicted a macroscopic signal visible to functional magnetic resonance imaging (fMRI) in humans. We then looked for this signal as participants explored a virtual reality environment, mimicking the rats' foraging task: fMRI activation and adaptation showing a speed-modulated six-fold rotational symmetry in running direction. The signal was found in a network of entorhinal/subicular, posterior and medial parietal, lateral temporal and medial prefrontal areas. The effect was strongest in right entorhinal cortex, and the coherence of the directional signal across entorhinal cortex correlated with spatial memory performance. Our study illustrates the potential power of combining single-unit electrophysiology with fMRI in systems neuroscience. Our results provide evidence for grid-cell-like representations in humans, and implicate a specific type of neural representation in a network of regions which supports spatial cognition and also autobiographical memory.

Many-body instability of Coulomb interacting bilayer graphene: Renormalization group approach

Low-energy electronic structure of (unbiased and undoped) bilayer graphene consists of two Fermi points with quadratic dispersions if trigonal warping is ignored. We show that short-range (or screened Coulomb) interactions are marginally relevant and use renormalization group to study their effects on low-energy properties of the system. We find that the two quadratic Fermi points spontaneously split into four Dirac points. This results in a nematic state that spontaneously breaks the sixfold lattice rotation symmetry (combined with layer permutation) down to a twofold one, with a finite transition temperature. Critical properties of the transition and effects of trigonal warping are also discussed.

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. radic7-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.

Two-dimensional photonic crystals with anisotropic unit cells imprinted from poly(dimethylsiloxane) membranes under elastic deformation

We study structural symmetries of two-dimensional (2D) photonic crystals with anisotropic unit cells, including square- and rectangular-lattices with orientationally modulated elliptic motifs, and a compound structure consisting of circles with sixfold rotational symmetry and elliptical lines with twofold symmetry, which are created through elastic deformation of a single elastomeric membrane with circular pores. We then investigate the photonic bandgap (PBG) properties of the corresponding 2D Si posts and their tolerance to the structural deviation. We find that in the compound structure the overall PBGs are dominated by the sublattice with a higher symmetry, while the total symmetry is determined by the one with a lower symmetry.

A transmission electron microscope and electron diffraction study of carbon nanodisks

The pyrolytic Kvaerner carbon-black and hydrogen process yields industrial amounts of carbon nanodisks and nanocones. We here report on the first detailed transmission electron microscope and electron diffraction study of such carbon nanodisks. The carbon nanodisk rim thickness ranges from 5–6 nm to 60–70 nm, but most disks are 10–30 nm thick. In agreement with earlier reported results, we find that the disk diameters fall within the range from 500 to 4000 nm. Most nanodisks display six identical pairs of facets. The two associated interfacial angles are θ 1 = (22 ± 1)° and θ 2 = 60.0°− θ 1. The nanodisk fracture surfaces reveal that the nanodisks are multi-layered structures. When the incident electron beam is perpendicular to the plane of the nanodisk we find that the diffraction patterns consist of concentric and continuous rings, on top of which there are discrete sets of diffraction spots. These spots display six-fold rotational symmetry and are consistent with the diffraction patterns of graphite layers.

Anisotropic elasticity in a textured cubic film plane

In textured films, the elastic properties are not isotropic but depend on a specific direction in question. Four expressions of Young's modulus E, Poisson's ratio υ, the biaxial elastic modulus M and Rigidity modulus G along an arbitrary direction M 1 in the film plane are given directly in terms of the Miller indices of the plane ( h k l) for cubic structure materials. These planar elastic anisotropies depend not only on the orientation of the plane but also on the value of the compliance anisotropy factor s 0= s 11– s 12–1/2 s 44. Practical calculations show that only in the (1 1 1)-textured film, the elastic properties are planar isotropic due to the (1 1 1) plane having the highest six-fold rotation symmetry. For a (1 0 0)-textured film, neither Young's modulus E nor Poisson's ratio υ alone are isotropic, but the educed biaxial elastic modulus M=( E/1− υ) is isotropic.

Growth of InN films on spinel substrates by pulsed laser deposition

AbstractWe have grown InN films on MgAl2O4(111) substrates with atomically flat surfaces using pulsed laser deposition (PLD) and compared their structural properties with those grown on (Mn,Zn)Fe2O4(111) substrates. It has been revealed that InN(0001) films grow on MgAl2O4(111) with an in‐plane epitaxial relationship of InN[1$ \bar 1 $00] // MgAl2O4[1$ \bar 1 $0], achieving a lattice mismatch minimum. The InN films exhibited a clear sixfold rotational symmetry, without 30° rotational domains and with a full width at half maximum value of the InN 0002 rocking curve being 17.5 arcmin. Comparison between InN films grown on MgAl2O4 and those on (Mn,Zn)Fe2O4 led us to conclude that suppression of the interfacial reactions between the InN films and the substrate is inherently important to obtain high quality InN on substrates with a spinel structure. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dependence of the stresses on grain orientations in hexagonal films

A thin polycrystalline film attached tightly to a thick substrate of different thermal expansion coefficients will experience thermal stresses when the temperature is changed during device fabrication and in service. Calculations of these stresses in various ( h k l )-oriented grains relative to the film surface have been made for a polycrystalline film composed of the hexagonal metal Be, Cd, Co, Hf, Mg, Re, Ru, Sc, Ti, Y, Zr and Zn, respectively. For all these hexagonal films, the stresses σ 1 and σ 2 in plane of the film surface are equal only in (0 0 1)-oriented grains due to the highest six-fold rotation symmetry of the crystallographic Z-axis. Excepting σ 1 of Be, Ru, Zr, Zn and σ 2 of Cd, Zn, the maximum values of the film plane stresses σ 1 and σ 2 correspond to the (0 0 1)-oriented grains means that the significant reliability problems, such as, voiding, cracking, hillocking induced by the stresses may be taken place preferred in (0 0 1)-oriented grains.

Effect of the subwavelength hole symmetry on the enhanced optical transmission through metallic films

We present here the measurements on the extraordinary optical transmission when light passes through different subwavelength hole arrays fabricated on a 120-nm-thick gold film by the focused ion beam system. The comparison of optical transmission shows that the aperiodic concentric circular hole arrays, which possess the sixfold rotational symmetry and central-inversion symmetry, have higher transmittance (about six and two times larger in peak intensity, respectively) than the periodic hexagonal and square lattice of hole array structures. This phenomenon can be attributed to the stronger constructive interference of multiple scattered waves in a higher symmetrical hole array structure. In addition, although the three structures possess the same lattice constant 1μm and hole diameter 350nm, the enhanced transmission peak for the circular hole arrays locates in the visible region, while that of the two periodic structures is in the infrared region. Moreover, when the hole diameter decreases, the transmission peak shifts slightly, while the transmittance drops rapidly.