One-dimensional Incommensurate Research Articles

Plane wave study on the localized-extended transition in the one-dimensional incommensurate systems

Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials, which are in close connection to many recent experiments of the ultracold atoms and photonic crystals. With the aid of numerical calculations, we formulate a propagator based scattering picture to explain the transition at the ground state and single particle mobility edge, in which deeper connections among the incommensurate potentials, eigenstate compositions and transition mechanism are revealed. We further show that there exists an upper limit of localization length for all localized eigenstates, leading to a fundamental difference to the Anderson localization. The scattering picture provides unique insights on the localization mechanism in the incommensurate systems, and can be generalized to study the more complex phenomena in higher dimensions, with different potentials or beyond the single particle regime.

A study of one-dimensional incommensurate modulated structure determination in high-resolution transmission electron microscopy

The methods for determining incommensurate modulated structures (IMSs) in high-resolution transmission electron microscopy have been studied and improved to a level more perfect than before. This is demonstrated by means of the IMS determination for Bi2.31Sr1.69CuO6+δas an example. First, as previously, the projected potential map (PPM) of the IMS with resolution 0.2 nm was obtained after image deconvolution from a [100] image. Secondly, the resolution of the PPM was enhanced to 0.1 nm through phase extension combined with the electron-diffraction data so that the substitutional and displacive modulation functions could be determined. Thirdly, a (2+1)-dimensional structure model that corresponds to the [100] projected IMS was built for calculating the related partial structure factors that were utilized to correct the experimental electron-diffraction intensities for both main and satellite reflections. After three cycles of diffraction-intensity correction and phase extension, all unoverlapped atoms projected along the [100] direction in Bi2.31Sr1.69CuO6+δwere resolved, and the modulations of substitution and displacement could be observed clearly. The substitution of Bi for Sr atoms at the Sr(O) columns was seen in the final PPM and verified by high-dimensional image simulation.

Topological Superconductor to Anderson Localization Transition in One-Dimensional Incommensurate Lattices

We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z2 invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system.

Localization of Bose-Fermi Mixtures in One-Dimensional Incommensurate Lattices

We studied the localization property of Bose-Fermi mixture systems on incommensurate optical lattices by using quantum Monte-Carlo simulations. We found a characteristic behavior of the superfluid density when changing the Bose-Bose interactions and Bose-Fermi interactions over a wide range. The superfluidity of the bosons was enhanced in the presence of the fermions by the repulsive Bose-Fermi interactions which counteracted the localization of the bosons. With attractive Bose-Fermi interactions, on the other hand, the superfluidity is also enhanced by a different mechanism when the number of the bosons is much larger than that of the fermions.

Predicted Mobility Edges in One-Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization

Localization properties of noninteracting quantum particles in one-dimensional incommensurate lattices are investigated with an exponential short-range hopping that is beyond the minimal nearest-neighbor tight-binding model. Energy dependent mobility edges are analytically predicted in this model and verified with numerical calculations. The results are then mapped to the continuum Schrödinger equation, and an approximate analytical expression for the localization phase diagram and the energy dependent mobility edges in the ground band is obtained.

Image Deconvolution - An Effective Tool of Crystal Structure and Defect Determination in High-Resolution Electron Microscopy

AbstractImage deconvolution is introduced as an effective tool to enhance the determination of crystal structures and defects in high-resolution electron microscopy. The essence is to transform a single image that does not intuitively represent the examined crystal structure into the structure image. The principle and method of image deconvolution together with the related image contrast theory, the pseudo weak phase object approximation (pseudo WPOA), are briefly described. The method has been applied to different types of dislocations, twin boundaries, stacking faults, and one-dimensional incommensurate modulated structures. Results on the semiconducting epilayers Si0.76Ge0.24/Si and 3C-SiC/Si are given in some detail. The results on other compounds including AlSb/GaAs, GaN, Y0.6Na0.4Ba2Cu2.7Zn0.3O7-δ, Ca0.28Ba0.72Nb2O6and Bi2.31Sr1.69CuO6+δare briefly summarized. It is also shown how to recognize atoms of Si from C based on the pseudo WPOA, when the defect structures in SiC was determined at the atomic level with a 200 kV LaB6microscope.

Adiabaticity and Localization in One-Dimensional Incommensurate Lattices

We experimentally investigate the role of localization on the adiabaticity of loading a Bose-Einstein condensate into a one-dimensional optical potential comprised of a shallow primary lattice plus one or two perturbing lattice(s) of incommensurate period. We find that even a very weak perturbation causes dramatic changes in the momentum distribution and makes adiabatic loading of the combined lattice much more difficult than for a single period lattice. We interpret our results using a band-structure model and the one-dimensional Gross-Pitaevskii equation.

Structure and electronic properties of gold adsorbed on Ti(0 0 0 1)

The Au/Ti(0 0 0 1) adsorption system was studied by low energy electron diffraction (LEED) and photoemission spectroscopy with synchrotron radiation after step-wise Au evaporation onto the Ti(0 0 0 1) surface. For adsorption of Au at 300 K, no additional superstructures were observed and the (1 × 1) pattern of the clean surface simply became diffuse. Annealing of gold layers more than 1 ML thick resulted in the formation of an ordered Au–Ti surface alloy. Depending on the temperature and annealing time, three surface reconstructions were observed by LEED: (√3 × √3) R30°, (2 × 2) and a one-dimensional incommensurate (√3 × √3) rectangular pattern. The Au 4f core level and valence band photoemission spectra provided evidence of a strong chemical interaction between gold and titanium. The data indicated formation of an intermetallic interface and associated valence orbital hybridization, together with diffusion of gold into the bulk. Au core-level shifts were found to be dependent on the surface alloy stoichiometry.

TEM study of superstructure in a perovskite lead lanthanum zirconate stannate titanate ceramic

The superstructure of an antiferroelectric Pb 0.97La 0.02(Zr 0.66Sn 0.25Ti 0.09)O 3 phase, whose composition is near the morphotropic-phase boundary, was characterized. Systematic selected-area diffraction revealed that there were two kinds of superlattice reflections in the pseudocubic reciprocal lattice, i.e. 1 2 hk l superlattice reflections ( h, k, l all odd), and g± 1 7.24 a ∗ +b ∗ one-dimensional incommensurate superlattice reflections, where g denotes the vectors of fundamental or 1 2 hk l superlattice reflections. Convergent-beam electron diffraction disclosed that the average structure of the phase was rhombohedral with space group of R 3m . Based on the rhombohedral reciprocal lattice, the 1 2 hk l reflections were no longer superlattice but fundamental reflections, and the reciprocal vector of the one-dimensional incommensurate reflections was re-expressed as H=ha ∗ +kb ∗ +lc ∗ ± 1 7.24 a ∗ +b ∗ , where h, k, l are integers and (− h + k + l) = 3 n. In the light of the average structure and the reflection condition, the superspace Bravais class of the phase with one-dimensional incommensurate structure was determined to be P 11 R 3m in a (3 + 1)-dimensional space. In addition, the origins of the superlattice reflections were also examined and discussed.

Incommensurate composite structure of the superconductor Bi2Sr2CaCu2O8+δ

We have performed elastic neutron scattering experiments, using a high-quality single crystal of the high-${T}_{c}$ superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}.$ By a systematic comparison of intensities between superlattice reflections, we show that the one-dimensional incommensurate modulated model, usually used to interpret the structural characteristics, exhibits important failures. In contrast, in accordance with theoretical predictions [M. B. Walker and W. Que, Phys. Rev. B 45, 8085 (1992)], a composite structure made of two interpenetrating subsystems can alternatively describe the incommensurate structure and account for the hierarchy of the observed intensities.

Incommensurate modulated structure study of a Cu–Zn–Al–Zr phase

Zr was added to Cu–Zn–Al shape memory alloy as a grain refinement element. There are two new phases, Cu 50.2Zr 24.6Al 17.3Zn 7.9 (at%) ( Z 1 phase) and Cu 57.4Zr 20.4Zn 10.3Al 11.9 (at%) ( Z 2 phase), in the Cu–19.0Zn–13.1Al–1.1Mn–0.3Zr (at%) alloy. The structure of the Cu 57.4Zr 20.4Zn 10.3Al 11.9 (at%) ( Z 2) phase is studied in detail in the present paper. The results of electron diffraction and high-resolution electron microscopy (HREM) investigations indicated a one-dimensional incommensurate modulated structure in the Z 2 phase. The average structure of the phase is base-centred orthorhombic. The systematic reflection conditions associated with the main and satellite reflections demonstrate that the Bravais class of the Z 2 phase is P Cmmm 11 1 type in a (3+1) dimensional space for the incommensurate modulated structure. The substitution modulation effects of the Al and Zr elements are revealed by energy-filtered transmission electron microscopy (EFTEM). { 1 3 ̄ 2 ̄ } compound-type twins are observed in the Z 2 phase. The relationship between the Z 1 and Z 2 phases in the alloy is discussed.

Incommensurate Modulated Structure of “Pb”-1223 Determined by Combining High Resolution Electron Microscopy and Electron Diffraction

The one-dimensional incommensurate modulated structure of (Pb0.5Sr0.3Cu0.2)Sr-2(Ca0.6Sr0.4)(2)Cu3Oy has been determined. The low-resolution average structure was derived by image deconvolution. The image resolution was enhanced by phase extension based on a set of corrected electron diffraction intensities. Multi-dimensional direct methods were then used to solve the four-dimensional (4D) structure. The incommensurate modulated structure was finally obtained by cutting the 4D structure with the three-dimensional physical space. Occupation and/or displacement modulation has been observed for all atoms.

Combined Standard Uncertainties for Interatomic Distance in One-Dimensionally Modulated Crystals

A method is proposed that allows a variance estimation for interatomic distances in the case of a one-dimensional incommensurate (or commensurate) displacively modulated structure to be derived. This method takes into account the covariance terms in the variance-covariance matrix Σ of the atomic pair, resulting from true linear relations between some positional parameters. These linear relations, arising in the case of symmetry-related atoms or in the case of special positions in crystals of high symmetry (trigonal, tetragonal or hexagonal systems), are carefully scrutinized. The results of these linear relations on the Σ matrix are specified for any order of the Fourier expansion of the modulated displacements and some examples of matrices are given. A general formula written in matrix form is then derived for the bond-length variance. This method is then exemplified by different typical examples, showing the usefulness of such a calculation for the comparison of different results or for the discussion of the significance of some interatomic distance variations.

ELECTRON-PHONON COUPLING CLOSE TO A METAL-INSULATOR TRANSITION IN ONE DIMENSION

We consider a one-dimensional system of electrons interacting via a short-range repulsion and coupled to phonons close to the metal-insulator transition at half-filling. We argue that the metal-insulator transition can be described as a standard one-dimensional incommensurate to commensurate transition, even if the electronic system is coupled to the lattice distortion. By making use of known results for this transition, we prove that low-momentum phonons, with the inclusion of the 4kF (≃2π near half-filling) scattering, do not play any relevant role close to the metal-insulator transition, unless their coupling to the electrons is large in comparison with the other energy scales present in the problem. In other words the effective strength of the low-momentum transferred electron-phonon coupling does not increase close to the metal-insulator transition, even though the effective velocity of the mobile carriers is strongly diminished.

X-ray investigation of the incommensurate modulated structure of Bi2.08Sr1.84CuO6−δ

At room temperature the bismuth strontium cuprate Bi2.osSrl.8aCuO6 [a = 5.3791 (6), b = 5.3811 (9), c = 24.589 (3) A, fl = 89.93 (1) °, basic space group A2/a, superspace group pAZ/~, M, = 755.44, Dx = 7.05 g cm -3, V = 711.7 ,/k 3, Z = 4, Cu Ka radiation, = 1.541838/~, /z = 1203 cm-i, F(000) = 1278.24, T = 294 K] exhibits a one-dimensional incommensurate modulated structure with a modulation wavevector q* having components 0.2030 (2), 0, 0.467 (2). Both a displacive modul~ition wave, acting on all atoms and a density modulation wave, governing the occupancy probabilities of the Bi, Sr and apical 0(2) sites are involved in the crystal. 1117 unique hklm reflections with I_> 3o-(1) were used to refine the modulated structure. The final R factors (based on F) of main reflections (292), and first- and secondorder satellite reflections (563 and 262) are 0.061, 0.061 and 0.086 respectively. The modulation is anharmonic for all atoms. Evolution of the Bi, Sr and apical 0(2) site occupancies throughout the crystal is correlated in a spectacular way and shows the existence of Bi and 0(2) vacant sites in some unit cells; moreover, some Sr sites are occupied by Bi atoms. These features give rise to various coordinations i.e. square planar, pyramidal, octahedral for

La1.16Mo8O16: a hollandite-related compound with an incommensurate modulated structure

The hollandite-related structure La1.16Mo8O16, Mr = 1184.66, tetragonal, P^{I4}_{\kern3pt 1}, a = 9.983 (1), c = 2.8890 (5) Å, V = 287.9 Å3, Z = 1, Dx = 6.83 g cm−3, \lambda(Mo Kα) = 0.71069 Å, μ = 126 cm−1, F(0000) = 530, room temperature, R = 0.041 for 1145 unique reflections with I\geq3σ(I). At room temperature the compound exhibits a one-dimensional incommensurate modulated structure with a modulation wavevector q* = 0.608 (1)c*. Both a displacive modulation wave, acting on La, Mo and O atoms, and a modulation wave governing the occupancy probability of La sites inside the tunnels are involved in the crystal. Within the superspace group P^{I4}_{\kern3pt 1}, the final R values of main reflections (477) and first- and second-order satellite reflections (523 and 145) are 0.030, 0.050 and 0.133 respectively. The more spectacular modulation features are the occurrence of La—La pairs in the tunnels and the formation of Mo3 triangular clusters in the double chains of edge-sharing octahedra. Contrary to previous descriptions based upon rigid tunnels in structures of the hollandite type, the tunnels in the crystal studied are distorted in a periodic way along [001] (λ = 4.75 Å), giving rise to alternate contractions and expansions. The distortion of the double octahedral chains is considerable and probably created via the La—O(1) bonds by insertion of La atoms inside two adjacent tunnels.

The dielectric tensor in incommensurately modulated crystal phases

A characterization of the dielectric properties in incommensurately modulated phases is presented. First a microscopic description is adopted. It takes into account the structural restrictions imposed on a microscopic scale by the superspace group. The macroscopic optical properties are derived from it. The optical activity as observed in the incommensurate phases of several compounds cannot be explained within a plain microscopic approximation. Therefore, a mesoscopic level is considered, implying averaging distances large with respect to the basic unit cell parameters or even the modulation wavelength, but small with respect to the size of the finite crystal. The phenomenological approach presented is compatible both with the microscopic description based on a superspace group symmetry, as well as with the macroscopic one involving the point group of the average structure. In the case considered here (one-dimensional incommensurate modulation) the optical properties of the dielectric medium are analysed in terms of subperiodic groups, periodic along the direction of the modulation wave and homogeneous along the other two directions. It is shown that the optical properties are affected by the incommensurability and that small changes in symmetry related to boundary conditions can lead to macroscopic effects. To interpret the experimental data more quantitatively a Jones model is applied, which describes the mesoscopic spatial dispersion of the modulated dielectric tensor to vary periodically along the modulation and being constant in the plane perpendicular to it. This model is compared with the experimental results for the ellipticity angle chi of (N(CH3)4)2ZnCl4 obtained for light propagating along the modulation.

Mechanisms of phase transitions between commensurate and incommensurate phases.

A three-dimensional model with one-dimensional incommensurate and commensurate phases has been simulated using the molecular-dynamics method. The results of simulation show that the commensurate\ensuremath{\rightarrow}incommensurate phase transition starts from nucleation of a stripple whose existence, in turn, promotes nucleation of the next stripples in its vicinity. That process repeats and leads to cascade nucleation of stripples, effectively regarded as a frontal motion of the commensurate-incommensurate interface. The mechanism of the phase transition between two incommensurate phases is different and relies, not on random nucleation of stripples, but on correlated nucleation of a series of stripples that all grow and propagate in one thin layer of the crystal, which is perpendicular to the modulation direction. Easy nucleation of stripples at some surfaces of the crystallite suppresses the serial nucleation effect. Stripple nucleations and growths are accompanied by heat release.

ELECTRON DIFFRACTION AND HIGH RESOLUTION MICRO-SCOPY STUDY ON INCOMMENSURATE MODULATED STRUCTURE IN Ce1+εFe4B4 ALLOY

The permanent magnetic alloy Ce1+εFe4B4(ε= 0.1, B-rich phase) is studied by means of electron diffraction and high resolution electron microscopy. It has been determined that the Ce1+εFe4B4 alloy has a one-dimensional incommensurate modulated structure of Chimney-Ladder type consisting of two overlapped substructures fomed by Ce atoms and Fe-B clusters respectively. The unit cell parameters of Ce substructure are a = 7.08?, cCe= 3.51-3.60?, and the possible space groups are I4,I4,I4mm, I4/m, I4/mmm. The unit cell parameters of Fe-B substructure are a = 7.08?, cFe= 3.91?, and the space group is P/42/mcm. All of the major and satellite diffraction spots are indexed by using of four indexes method. The high resolution electron microscopic image shows contrast modulation bands corresponding to the modulated structure. The fluctuation of the period of Ce substructure has been observed and discussed. The orientational anormaly caused by the possible deviation of Ce atomic planes from the (a, b) plane is also discused.

ELECTRON DIFFRACTION STUDY OF THE ONE DIMENSIONAL INCOMMENSURATE MODULATED STRUCTURE IN ANKANGITE

The one-dimensional incommensurate modulated structure of ankangite (Ba0.827(Ti5.827V2.294Cr0.053)O16), a mineral recently discovered in China, has been studied by electron diffraction. Its average structure belongs to tetragonal system and unit cell parameters of the average structure are a= 10.2?, c=2.96?. The modulated wave spreading along direction c has the wave length of λ = 2.30c. The features of electron diffraction patterns are analysed in detail. The possible mechanism of the incommensurate modulated structure in ankangite is discussed and a modulated structure model of vacancy-displacive type is proposed. The disorder structure caused by the difference among initial phases for displacement functions from place to place is discussed.