Twofold Rotation Symmetry Research Articles

Classification of accidental band crossings and emergent semimetals in two-dimensional noncentrosymmetric systems

We classify all possible gap-closing procedures which can be achieved in two-dimensional time-reversal invariant noncentrosymmetric systems. For exhaustive classification, we examine the space-group symmetries of all 49 layer groups lacking inversion, taking into account spin-orbit coupling. Although a direct transition between two insulators is generally predicted to occur when a band crossing happens at a general point in the Brillouin zone, we find that a variety of stable semimetal phases with point or line nodes can also arise due to the band crossing in the presence of additional crystalline symmetries. Through our theoretical study, we provide the complete list of nodal semimetals created by a band inversion in two-dimensional noncentrosymmetric systems with time-reversal invariance. The transition from an insulator to a nodal semimetal can be grouped into three classes depending on the crystalline symmetry. First, in systems with a twofold rotation about the $z$ axis (normal to the system), a band inversion at a generic point generates a two-dimensional Weyl semimetal with point nodes. Second, when the band crossing happens on the line invariant under a twofold rotation (mirror) symmetry with the rotation (normal) axis lying in the two-dimensional plane, a Weyl semimetal with point nodes can also be obtained. Finally, when the system has a mirror symmetry about the plane embracing the whole system, a semimetal with nodal lines can be created. Applying our theoretical framework, we identify various two-dimensional materials as candidate systems in which stable nodal semimetal phases can be induced via doping, applying electric field, or strain engineering, etc.

1,3-Bis(4-bromobutoxy)benzene

The whole molecule of the title compound, C14H20Br2O2, is generated by twofold rotational symmetry, with the twofold axis bisecting the benzene ring. The packing of the molecules features C—H...π interactions, which link the molecules to form chains along [100].

Structural characterization of the NAP; the major adhesion complex of the human pathogen Mycoplasma genitalium.

Mycoplasma genitalium, the causative agent of non-gonococcal urethritis and pelvic inflammatory disease in humans, is a small eubacterium that lacks a peptidoglycan cell wall. On the surface of its plasma membrane is the major surface adhesion complex, known as NAP that is essential for adhesion and gliding motility of the organism. Here, we have performed cryo-electron tomography of intact cells and detergent permeabilized M. genitalium cell aggregates, providing sub-tomogram averages of free and cell-attached NAPs respectively, revealing a tetrameric complex with two-fold rotational (C2) symmetry. Each NAP has two pairs of globular lobes (named α and β lobes), arranged as a dimer of heterodimers with each lobe connected by a stalk to the cell membrane. The β lobes are larger than the α lobes by 20%. Classification of NAPs showed that the complex can tilt with respect to the cell membrane. A protein complex containing exclusively the proteins P140 and P110, was purified from M. genitalium and was structurally characterized by negative-stain single particle EM reconstruction. The close structural similarity found between intact NAPs and the isolated P140/P110 complexes, shows that dimers of P140/P110 heterodimers are the only components of the extracellular region of intact NAPs in M. genitalium.

Weyl points and Dirac lines protected by multiple screw rotations

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron’s energy bands to have Weyl points at which two bands touch. This is illustrated for space groups No. 19 (P212121) and No. 198 (P213), which have three orthogonal screw rotation axes. In the case of space groups No. 61 (Pbca) and No. 205 (Pa-3) that have extra inversion symmetry, Weyl points are promoted to four-fold degenerate line nodes in glide-invariant planes. The three-fold rotation symmetry present in the space groups No. 198 and No. 205 allows Weyl and Dirac points, respectively, to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the Γ point and R point, respectively.

Crystal structure of a pyrazine-2,3-dicarboxamide ligand and of its silver(I) nitrate complex, a three-dimensional coordination polymer.

The title ligand, C18H16N6O2·2H2O (L1) [N2,N3-bis-(pyridin-4-ylmeth-yl)pyrazine-2,3-dicarboxamide], crystallized as a dihydrate. The mol-ecule is U-shaped with the carboxamide groups being cis to one another, making a dihedral angle of 81.6 (5)°. The terminal pyridine rings are inclined to one another by 58.5 (4)°. There is an intra-molecular N-H⋯Npyrazine hydrogen bond present, forming an S(5) ring motif. In the crystal, adjacent mol-ecules are linked by N-H⋯Ocarboxamide hydrogen bonds, forming a chain along [001]. A chain of hydrogen-bonded water mol-ecules is linked to the chain of (L1) mol-ecules by O-H⋯N hydrogen bonds, forming columns propagating along the c axis. The columns are linked by C-H⋯O and C-H⋯N hydrogen bonds, forming a three-dimensional supra-molecular structure. The reaction of ligand (L1) with silver(I) nitrate led to the formation of a new three-dimensional coordination polymer, {[Ag(C18H16N6O2)]NO3} n , poly[[[μ4-N2,N3-bis-(pyridin-4-ylmeth-yl)pyrazine-2,3-dicarboxamide]-silver(I)] nitrate] (I). The asymmetric unit is composed of half of one silver ion, located on a twofold rotation axis, half a ligand mol-ecule and half a positionally disordered nitrate anion located about a twofold rotation axis. The full mol-ecule of the ligand is generated by twofold rotational symmetry, with this twofold axis bis-ecting the Car-Car bonds of the pyrazine ring and the Ag-Ag bond. The carboxamide groups are now trans to one another, making a dihedral angle of 65.8 (4)°. The two terminal pyridine rings are inclined to one another by 6.6 (3)°. Two ligands wrap around an Ag-Ag bond of 3.1638 (11) Å, forming a figure-of-eight-shaped complex mol-ecule. Each silver ion is coordinated by two pyridine N atoms and by two carboxamide O atoms of neighbouring mol-ecules, hence forming a three-dimensional framework. The nitrate anion is linked to the framework by N-H⋯O and C-H⋯O hydrogen bonds.

Unconventional Topological Phase Transition in Two-Dimensional Systems with Space-Time Inversion Symmetry.

We study a topological phase transition between a normal insulator and a quantum spin Hall insulator in two-dimensional (2D) systems with time-reversal and twofold rotation symmetries. Contrary to the case of ordinary time-reversal invariant systems, where a direct transition between two insulators is generally predicted, we find that the topological phase transition in systems with an additional twofold rotation symmetry is mediated by an emergent stable 2D Weyl semimetal phase between two insulators. Here the central role is played by the so-called space-time inversion symmetry, the combination of time-reversal and twofold rotation symmetries, which guarantees the quantization of the Berry phase around a 2D Weyl point even in the presence of strong spin-orbit coupling. Pair creation and pair annihilation of Weyl points accompanying partner exchange between different pairs induces a jump of a 2D Z_{2} topological invariant leading to a topological phase transition. According to our theory, the topological phase transition in HgTe/CdTe quantum well structure is mediated by a stable 2D Weyl semimetal phase because the quantum well, lacking inversion symmetry intrinsically, has twofold rotation about the growth direction. Namely, the HgTe/CdTe quantum well can show 2D Weyl semimetallic behavior within a small but finite interval in the thickness of HgTe layers between a normal insulator and a quantum spin Hall insulator. We also propose that few-layer black phosphorus under perpendicular electric field is another candidate system to observe the unconventional topological phase transition mechanism accompanied by the emerging 2D Weyl semimetal phase protected by space-time inversion symmetry.

Metasurface polarization splitter.

Polarization beam splitters, devices that separate the two orthogonal polarizations of light into different propagation directions, are among the most ubiquitous optical elements. However, traditionally polarization splitters rely on bulky optical materials, while emerging optoelectronic and photonic circuits require compact, chip-scale polarization splitters. Here, we show that a rectangular lattice of cylindrical silicon Mie resonators functions as a polarization splitter, efficiently reflecting one polarization while transmitting the other. We show that the polarization splitting arises from the anisotropic permittivity and permeability of the metasurface due to the twofold rotational symmetry of the rectangular unit cell. The high polarization efficiency, low loss and low profile make these metasurface polarization splitters ideally suited for monolithic integration with optoelectronic and photonic circuits.This article is part of the themed issue 'New horizons for nanophotonics'.

{X,Y}-Cyclacene Graphs with Next Nearest Neighbor Interactions

ABSTRACTEigensolutions of {X( = C,B,N),Y( = C,B,N)}-cyclacene graphs with next nearest neighbor (nnn) interactions have been obtained in analytical forms by adapting n-fold rotational symmetry followed by two-fold rotational symmetry (or a plane of symmetry). Expressions of eigensolution indicate the subspectral relationship among such cyclacenes with an even number of hexagonal rings e.g., eigenvalues of {X,Y}-di-cyclacene are found in the eigenspectra of all such even cyclacenes. Total π-electron energies and highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO–LUMO) gaps are calculated using the analytical expressions obtained and are found to vary negligibly with the variation of nnn interactions in such cyclacenes. Total π-electron energy is found to increase due to increase in restriction intensity of nnn interactions, whereas the HOMO–LUMO gap of polyacenecs having the even number of hexagonal rings and with one electron at each site (atom) decreases with increase in the restriction intensity since such systems contain degenerate half-filled HOMO (bonding or nonbonding) that are much more vulnerable for perturbations imposed through nnn interactions.

Full spatial characteristics of emission from electron oscillation driven by linearly polarized laser pulses

Driven by the electromagnetic field produced by linearly polarized femtosecond laser pulse, the electron’s relativistic oscillation and the corresponding full spatial radiation have been studied theoretically and numerically. The new finding is electron’s radiated power per solid angle by laser pulse with low intensities is almost the same as that produced by a dipole antenna (four-fold rotational symmetry), whereas for the radiation generated by high laser intensities, the radiation takes on two-fold rotational symmetry pattern with further increase of laser intensity. Therefore, this method is of great use in determining the intensity of linearly polarized laser pulses indirectly in real experiment.

2,5-Bis[(3,3,4,4,5,5,6,6,7,7,8,8,8-tridecafluorooctyl)sulfanyl]-1,3,4-thiadiazole

The title compound, C18H8F26N2S3, was obtained by doubleS-perfluorohexylethylation of dipotassium 1,3,4-thiadiazole-2,5-dithiolate in methanol. The molecule exhibits twofold rotational symmetry, with the S atom lying on the rotation axis. The fluorocarbon chains adopt helical conformations and the F atoms of the two terminal C atoms are disordered over two sets of sites. No directional intermolecular interactions occur in the crystal.

In Situ Oxidation of GaN Layer and Its Effect on Structural Properties of Ga2O3 Films Grown by Plasma-Assisted Molecular Beam Epitaxy

Plasma-assisted molecular beam epitaxy (PAMBE) was used to grow Ga2O3 films on oxidized GaN layers on nitrided sapphire substrates. The GaN layer was grown by PAMBE, and the in situ oxidation of the GaN layer was achieved through exposure to oxygen plasma, which resulted in the formation of monoclinic β-Ga2O3. Crystalline monoclinic β-Ga2O3 films were grown on the GaN layers, with and without oxidation. The orientation relationships were [\(11\overline{2} 0\)] Al2O3//[\(1\overline{1} 00\)] AlN//[\(1\overline{1} 00\)] GaN//[102] β-Ga2O3 and [\(1\overline{1} 00\)] Al2O3//[\(11\overline{2} 0\)] AlN//[\(11\overline{2} 0\)] GaN//[010] β-Ga2O3. The grown β-Ga2O3 films were not single-crystalline but showed rotational domains along the growth direction with three variations, which resulted in six-fold rotational symmetry instead of two-fold rotational symmetry. The surface roughness of the grown β-Ga2O3 film was closely reflected to that of as-grown GaN and oxidized GaN. By analyzing the x-ray omega rocking curves for the on-axis (\(\overline{2} 01\)) and off-axis (002) reflections, it was concluded that rotational domains dominantly affected the crystal quality of the β-Ga2O3 films.

Crystal structure of bis-[cis-(1,4,8,11-tetra-aza-cyclo-tetra-deca-ne-κ4N)bis(thio-cyanato-κN)chrom-ium(III)] dichromate monohydrate from synchrotron X-ray diffraction data.

The structure of the complex salt, cis-[Cr(NCS)2(cyclam)]2[Cr2O7]·H2O (cyclam= 1,4,8,11-tetra-aza-cyclo-tetra-decane, C10H24N4), has been determined from synchrotron data. The asymmetric unit comprises of one [Cr(NCS)2(cyclam)]+ cation, one half of a Cr2O72- anion (completed by inversion symmetry) and one half of a water mol-ecule (completed by twofold rotation symmetry). The CrIII ion is coordinated by the four cyclam N atoms and by two N atoms of cis-arranged thio-cyanate anions, displaying a distorted octa-hedral coordination sphere. The Cr-N(cyclam) bond lengths are in the range 2.080 (2) to 2.097 (2) Å while the average Cr-N(NCS) bond length is 1.985 (4) Å. The macrocyclic cyclam moiety adopts the cis-V conformation. The bridging O atom of the dichromate anion is disordered around an inversion centre, leading to a bending of the Cr-O-Cr bridging angle [157.7 (3)°]; the anion has a staggered conformation. The crystal structure is stabilized by inter-molecular hydrogen bonds involving the cyclam N-H groups and water O-H groups as donor groups, and the O atoms of the Cr2O72- anion and water mol-ecules as acceptor groups, giving rise to a three-dimensional network.

Anisotropic non-Fermi liquids

We study non-Fermi liquid states that arise at the quantum critical points associated with the spin density wave (SDW) and charge density wave (CDW) transitions in metals with twofold rotational symmetry. We use the dimensional regularization scheme, where a one-dimensional Fermi surface is embedded in $3-\epsilon$ dimensional momentum space. In three dimensions, quasilocal marginal Fermi liquids arise both at the SDW and CDW critical points : the speed of the collective mode along the ordering wavevector is logarithmically renormalized to zero compared to that of Fermi velocity. Below three dimensions, however, the SDW and CDW critical points exhibit drastically different behaviors. At the SDW critical point, a stable anisotropic non-Fermi liquid state is realized for small $\epsilon$, where not only time but also different spatial coordinates develop distinct anomalous dimensions. The non-Fermi liquid exhibits an emergent algebraic nesting as the patches of Fermi surface are deformed into a universal power-law shape near the hot spots. Due to the anisotropic scaling, the energy of incoherent spin fluctuations disperse with different power laws in different momentum directions. At the CDW critical point, on the other hand, the perturbative expansion breaks down immediately below three dimensions as the interaction renormalizes the speed of charge fluctuations to zero within a finite renormalization group scale through a two-loop effect. The difference originates from the fact that the vertex correction anti-screens the coupling at the SDW critical point whereas it screens at the CDW critical point.

Diethyl 2,2′-(trisulfane-1,3-diyl)dibenzoate

The title compound, C18H18O4S3, was synthesized in the presence of chloridoauric acid from 3H-1,2-benzodithiole-3-thione as starting material. The asymmetric unit comprises one half of the molecule, the complete molecule being generated by the application of twofold rotation symmetry. The two benzene rings are inclined by 81.0 (2)°. In the crystal, slipped π–π stacking interactions are observed between the benzene rings of neighbouring molecules.

Crystal structure of 1-phenyl-imido-1-{6-[1-(phenyl-imino)-eth-yl]pyridin-2-yl}ethan-1-yl-κ3N,N',N'')iron(II).

The title iron complex, [Fe(C21H19N3)2], consists of an FeII atom chelated by two tridentate bis-(imino)-pyridine radical anions in a slightly distorted octa-hedral coordination environment. In the solid state, there are two independent half-mol-ecules in the asymmetric unit, and the complete mol-ecular structure is formed by applying twofold rotation symmetry with the twofold rotation axis passing through an Fe atom. In the crystal, the Fe-containing complexes are not involved in any particular direct inter-molecular inter-actions, with the shortest C-HAr contacts between neighboring phenyl groups being ca 3.2 Å.

Chloridobis[2-(pyridin-2-yl-κN)benzo[b][1,5]naphthyridine-κN 1]copper(II) perchlorate acetonitrile disolvate

The copper(II) ion in the title complex, [CuCl(C17H11N3)2]ClO4·2CH3CN, is coordinated by four N atoms from two pbn ligands and one Cl− ion in a distorted trigonal–bipyramidal geometry (τ = 0.84). The asymmetric unit comprises half of the cationic complex molecule, and complete molecules are generated by twofold rotation symmetry with the corresponding axis running through the Cu atom and the coordinating Cl atom. The perchlorate anion is also located on a twofold rotation axis (passing through the Cl atom). In the crystal, there are π–π stacking interactions between the benzonaphthyridine rings of the pbn ligand of neighbouring cations.

5,6-Bis(pyridin-2-yl)-2,3-dihydropyrazine

The title compound, C14H12N4, has approximate twofold rotational symmetry. The pseudo-twofold axis bisects the C—C bonds of the dihydropyrazine ring, which has a screw–boat conformation. The two pyridine rings are inclined to the mean plane of the dihydropyrazine ring by 30.78 (11) and 39.37 (9)°, and to one another by 62.53 (10)°. The pyridine N atoms arecisto one another, with an N...N nonbonded distance of 3.101 (2) Å. In the crystal, molecules are linkedviaa pair of N—H...N hydrogen bonds, forming inversion dimers with anR22(6) ring motif. These units are linked by further N—H...H hydrogen bonds, forming layers parallel to (302). The layers are linked by C—H...π interactions, forming a three-dimensional framework.

Crystal structure of cis-di-chlorido-(1,4,8,11-tetra-aza-cyclo-tetra-decane-κ4N)chromium(III) (oxalato-κ2O1,O2)(1,4,8,11-tetra-aza-cyclo-tetra-decane-κ4N)chromium(III) bis(perchlorate) from synchrotron data.

In the asymmetric unit of the title compound, [CrCl2(C10H24N4)][Cr(C2O4)(C10H24N4)](ClO4)2 (C10H24N4 = 1,4,8,11-tetra-aza-cyclo-tetra-decane, cyclam; C2O4 = oxalate, ox), there are two independent halves of the [CrCl2(cyclam)]+ and [Cr(ox)(cyclam)]+ cations, and one perchlorate anion. In the complex cations, which are completed by application of twofold rotation symmetry, the CrIII ions are coordinated by the four N atoms of a cyclam ligand, and by two chloride ions or one oxalate bidentate ligand in a cis arrangement, displaying an overall distorted octa-hedral coordination environment. The Cr-N(cyclam) bond lengths are in the range of 2.075 (5) to 2.096 (4) Å while the Cr-Cl and Cr-O(ox) bond lengths are 2.3358 (14) and 1.956 (4) Å, respectively. Both cyclam moieties adopt the cis-V conformation. The slightly distorted tetra-hedral ClO4- anion remains outside the coordination sphere. The supra-molecular architecture includes N-H⋯O and N-H⋯Cl hydrogen bonding between cyclam NH donor groups, O atoms of the oxalate ligand or ClO4- anions and one Cl ligand as acceptors, leading to a three-dimensional network structure.

Observation of Ising spin-nematic order and its close relationship to the superconductivity in FeSe single crystals

Superconducting FeSe single crystals of (001) orientation are synthesized via a hydrothermal ion-release route. An Ising spin-nematic order is identified by our systematic measurements of in-plane angular-dependent magnetoresistance (AMR) and static magnetization. The turn-on temperature of anisotropic AMR signifies the Ising spin-nematic ordering temperature Tsn, below which a two-fold rotational symmetry is observed in the iron plane. A downward curvature appears below Tsn in the temperature dependence of static magnetization for the weak in-plane magnetic field as reported previously. Remarkably, we find a universal linear relationship between Tc and Tsn among various superconducting samples, indicating that the spin nematicity and the superconductivity in FeSe have a common microscopic origin.

Evidence for Eight-Node Mixed-Symmetry Superconductivity in a Correlated Organic Metal.

We report on a combined theoretical and experimental investigation of the superconducting state in the quasi-two-dimensional organic superconductor κ-(ET)_{2}Cu[N(CN)_{2}]Br. Applying spin-fluctuation theory to a low-energy, material-specific Hamiltonian derived from abinitio density functional theory we calculate the quasiparticle density of states in the superconducting state. We find a distinct three-peak structure that results from a strongly anisotropic mixed-symmetry superconducting gap with eight nodes and twofold rotational symmetry. This theoretical prediction is supported by low-temperature scanning tunneling spectroscopy on insitu cleaved single crystals of κ-(ET)_{2}Cu[N(CN)_{2}]Br with the tunneling direction parallel to the layered structure.