Bloch Wave Method Research Articles

Tunneling in Cosine Potential with Periodic Boundary Conditions

In this paper we discuss three methods to calculate energy splitting in cosine potential on a circle, Bloch waves, semi--classical approximation and restricted basis approach. While the Bloch wave method gives only a qualitative result, with the WKB method we are able to determine its unknown coefficients. The numerical approach is most exact and enables us to extract further corrections to previous results.

Calculation of the emission depth distribution function for electrons emitted from a crystal by the Bloch wave method

The quantum mechanical Bloch-wave method combined with the reciprocity principle has been used to describe electron transport in crystalline solids for determining the emission depth distribution function (EMDDF) of signal electrons in surface electron spectroscopy. The crystal structure and electron coherent scattering are explicitly taken into account; these are neglected in classical Monte Carlo methods. While electron inelastic scattering is treated with an optical potential, the EMDDF for Auger-electron emission from an Au (001) surface is calculated by the quantum method. The dependence of the EMDDF on emission angle and crystal structure has indicated that this quantum EMDDF strongly depends on the emission direction of Auger electrons and is sensitive to crystal structure. Comparisons made between the present results and classical Monte Carlo simulation results show pronounced differences due to electron-diffraction effects. The mean escape depth (MED) which describes the surface sensitivity of Auger electron spectroscopy, is also evaluated; the present calculation shows that the MED for a crystalline solid is about half that for an amorphous solid from a Monte Carlo calculation.

On a simple scheme for computing the electronic energy levels of a finite system from thoseof the corresponding infinite system

Computing the electronic energy levels of a finite system or nanostructure is more difficultthan computing those of an infinite system or bulk material. In the literature, a techniquefor simplifying this computation has been proposed, wherein energy levels of a finite systemare derived from those of the corresponding infinite system. So far, this method has beenvalidated only for finite length one-dimensional systems and for higher-dimensional systems atk = 0. We establish that this technique, hereafter referred to as the confined Bloch wave (CBW)method, is valid for higher-dimensional symmorphic systems over the entire Brillouin zone,provided some symmetry requirements are satisfied. For this purpose we use alateral surface superlattice as a model for the infinite system and a stripe orribbon patterned in this superlattice as a model for the nanostructure. Finally, wecompute the subbands of zigzag ribbons of one type patterned in artificial grapheneand show that the CBW method predicts all the important subbands in theseribbons, and provides additional insight into the nature of their wavefunctions.

STEM image simulation by Bloch-wave method with layer-by-layer representation

In a Bloch-wave-based STEM image simulation, a framework for calculating the cross section for any incoherent scattering process was formulated by Allen et al. [(2003) Lattice-resolution contrast from a focused coherent electron probe. Part I. Ultramicroscopy 96: 47-63; Part II. ibid. 96: 65-81]. They simulated the high-angle annular dark-field, back-scattered electron, electron energy-loss spectroscopy and energy-dispersive X-ray (EDX) STEM images from the inelastic scattering coefficients. Furthermore, a skilful approach for deriving the excitation amplitude and block diagonalization in the eigenvalue equation was employed to reduce computing time and memory. In the present work, I extended their scheme to a layer-by-layer representation for application to inhomogeneous crystals. Calculations for a multi-layer Si sample including a displaced layer were performed by multiplying Allen et al.'s block-diagonalized matrices. Electron intensities within the sample and EDX STEM images were calculated at various conditions. From the calculations, three-dimensional STEM analysis was considered.

Essential experimental parameters for quantitative structure analysis using spherical aberration-corrected HAADF-STEM

The accuracy of quantitative analysis for Z-contrast images with a spherical aberration ( C s) corrected high-angle annular dark-field (HAADF) scanning transmission electron microscope (STEM) using SrTiO 3(0 0 1) was systematically investigated. Atomic column and background intensities were measured accurately from the experimental HAADF-STEM images obtained under exact experimental condition. We examined atomic intensity ratio dependence on experimental conditions such as defocus, convergent semi-angles, specimen thicknesses and digitalized STEM image acquisition system: brightness and contrast. In order to carry out quantitative analysis of C s-corrected HAADF-STEM, it is essential to determine defocus, to measure specimen thickness and to fix setting of brightness, contrast and probe current. To confirm the validity and accuracy of the experimental results, we compared experimental and HAADF-STEM calculations based on the Bloch wave method.

ALCHEMI study of chromium doped iron-aluminides

Abstract Mechanical properties of intermetallic iron-aluminides are strongly influenced by thermal pretreatment, stoichiometry and additional alloying elements. In the present study samples of FeAl and FeAl alloyed with Cr annealed at different temperatures were investigated. A revised ALCHEMI technique (Atom Location by CHanneling Enhanced MIcroanalysis) was employed to study site occupancies. The orientation dependences of characteristic X-ray yields were measured. Dynamic electron diffraction conditions with the systematic row excitation (planar ALCHEMI) were used. Experimental data were fitted by a theoretical calculation based on Bloch wave method. The distribution of elements on individual sublattices was obtained and preferential occupation of Al sites by Cr was confirmed.

Bloch-wave-based STEM image simulation with layer-by-layer representation

In a dynamical STEM image simulation by the Bloch-wave method, Allen et al. formulated a framework for calculating the cross-section for any incoherent scattering process from the inelastic scattering coefficients: thermal diffuse scattering (TDS) for high-angle annular dark-field (HAADF) and back-scattered electron (BSE) STEM, and ionization for electron energy-loss spectroscopy (EELS) and energy-dispersive X-ray spectroscopy (EDX) STEM. Furthermore, their method employed a skilful approach for deriving the excitation amplitude and block diagonalization in the eigenvalue equation. In the present work, we extend their scheme to a layer-by-layer representation for application to inhomogeneous crystals that include precipitates, defects and atomic displacement. Calculations for a multi-layer sample of Si–Sb–Si were performed by multiplying Allen et al.'s block-diagonalized matrices. Electron intensities within the sample and EDX STEM images, as an example of the inelastic scattering, were calculated at various conditions. From the calculations, 3-dimensional STEM analysis was considered.

Crystal structure refinement from electron diffraction data

A procedure of crystal structure refinement from electron diffraction data is described. The electron diffraction data on polycrystalline films are processed taking into account possible overlap of reflections and two-beam interaction. The diffraction from individual single crystals in an electron microscope equipped with a precession attachment is described using the Bloch-wave method, which takes into account multibeam scat- tering, and a special approach taking into consideration the specific features of the diffraction geometry in the precession technique. Investigations were performed on LiF, NaF, CaF 2 , and Si crystals. A method for reducing experimental data, which allows joint electron and X-ray diffraction study, is proposed.

Bloch wave homogenization of a non-homogeneous Neumann problem

In this paper, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the size of the holes goes to zero more rapidly than the domain period. This method allows to prove that, when the hole size exceeds a given threshold, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as “the strange term” in the literature.

ASTRA– a program package for accurate structure analysis by the intermeasurement minimization method

ASTRA2.0is a program package for single-crystal structure research using different experimental conditions and radiation of different types (X-rays, neutrons and electrons). The package includes programs for data collection optimization, data reduction, structural parameter refinement and electron density plotting. The expression that is responsible for minimization of the difference between reduced measurements is included in the goal function of the refinement program. The refinement program is based on an adaptive nonlinear least-squares method. The crystal structure is described by a detailed model for electron density analysis.ASTRAuses a multipole model up to eighth order for the aspherical electron density description, an anharmonic model up to sixth order for the description of atom displacement and a model of mixed atomic positions for the description of atomic replacement. Refinement of the structural parameters using data of precession electron diffraction and the Bloch-wave method to account for multiwave scattering is realized. Tools for refinement automation are included inASTRA.

Bloch wave homogenization in a medium perforated by critical holes

In this Note, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the hole size goes to zero more rapidly than the domain period. We prove that for a critical size, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as ‘the strange term’ in the literature. To cite this article: J. Ortega et al., C. R. Mecanique 335 (2007).

Crystal structure refinement using Bloch-wave method for precession electron diffraction

Procedure for crystal structure refinement using precession electron diffraction data and Bloch-wave method for accounting multibeam scattering is described. Refinement procedure takes into account features of precession geometry. Refining model consists of structural and reduced parameters determining dynamic diffraction. Difference between measured and calculated dynamic intensities of reflections is minimized with application of a nonlinear least squares method. As test example we used Si single nanocrystals. The influence of the reduced parameters on the quality of the obtained model is discussed. Refinement procedure is a part of ASTRA software.

Extended dynamical HAADF STEM image simulation using the Bloch-wave method

An extended method is proposed for the precise simulation of high-angle annular dark-field (HAADF) scanning transmission electron-microscope (STEM) images for materials containing elements with large atomic numbers and for thick specimens. The approach combines a previously reported method utilizing two kinds of optical potential [Watanabe, Yamazaki, Hashimoto & Shiojiri (2001). Phys. Rev. B, 64, 115432] with a representation of a crystal sliced into multiple layers. The validity of the method is demonstrated by simulated images for elements with the diamond structure (Si, Ge and alpha-Sn) and for the perovskite BaTiO3.

Bloch wave homogenization of linear elasticity system

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process.

HRTEM simulation in determination of thickness and grain misorientation for hydroxyapatite crystals

High-resolution transmission electron microscopy (HRTEM) and HRTEM simulation by the Bloch wave method (JEMS) are used to determine the structure and thickness of micro-and nanocrystals of biominerals and hydroxyapatite grown from aqueous solutions. It is established that thin (from one to several lattice parameters) crystals, including hydroxyapatite in mineralized biological tissues, are usually formed in low-temperature (up to 40°C) solutions. Relatively thick (up to several tens of lattice parameters) crystals grow only in high-temperature (~95°C) aqueous solutions. HRTEM simulation showed that crystals with a thickness exceeding one lattice parameter consist of nanograins misoriented with respect to one another along various directions within an angle of 0.7°.

A Combination Method of Charge Density Measurement in Hard Materials Using Accurate, Quantitative Electron and X-ray Diffraction: The α-Al2O3 Case

Current X-ray diffraction techniques intended for "ideally imperfect" specimens provide structure factors only on a relative scale and ever-present multiple scattering in strong low-angle Bragg reflections is difficult to correct. Multiple scattering is implicit in the quantitative convergent beam electron diffraction (QCBED) method, which provides absolutely scaled structure factors. Conventional single crystal X-ray diffraction has proved adequate in softer materials where crystal perfection is limited. In hard materials, the highly perfect nature of the crystals is often a difficulty, due to the inadequacy of the conventional corrections for multiple scattering (extinction corrections). The present study on alpha-Al2O3 exploits the complementarity of synchrotron X-ray measurements for weak and medium intensities and QCBED measurement of the strong low-angle reflections. Two-dimensional near zone axis QCBED data from different crystals at various accelerating voltages, thicknesses, and orientations have been matched using Bloch-wave and multislice methods. The reproducibility of QCBED data is better than 0.5%. The low-angle strong QCBED structure factors were combined with middle and high-angle extinction-free data from synchrotron X-ray diffraction measurements. Static deformation charge density maps for alpha-Al2O3 have been calculated from a multipole expansion model refined using the combined QCBED and X-ray data.

Structure and bonding in alpha-copper phthalocyanine by electron diffraction.

Energy-filtered quantitative electron diffraction at liquid nitrogen temperature has been used to examine the atomic structure and bonding of metastable alpha-Cu phthalocyanine crystals. Three theoretical methods (kinematic, kinematic with excitation errors and Bloch wave) were employed for the intensity calculations. The Bloch-wave method was found to account for dynamical effects by greatly reducing the residual factor between experimental and simulated results. A new method for calculating electron scattering factors for partially charged ions is proposed and the sensitivity of electron diffraction to charge transfer is discussed. The atomic charge states were analyzed for alpha-Cu phthalocyanine using a charge cloud model in which the Gaussian bond charge is positioned along the bonds. Spot patterns were collected in the Kohler mode at two beam energies to reduce error. Using the best-fitting model, a deformation charge-density map is produced and compared to the neutral-atom model. From this, the main features of atomic charge transfer in the alpha-Cu phthalocyanine structure can be seen in the (010) plane.

Lattice-resolution contrast from a focused coherent electron probe. Part II

In the previous paper, boundary conditions matching the probe to the crystal wave function in scanning transmission electron microscopy were applied by matching the whole wave function across the boundary. It is shown here how that approach relates to previous Bloch wave formulations using (phase-linked) plane wave boundary conditions for wave vectors implied by the range of transverse momentum components in the incident probe. Matching the whole wave function across the boundary, and including a suitably fine mesh in the reciprocal space associated with the crystal to allow matching of transverse momentum components within the probe, leads to a structure matrix A containing many elements which would normally be excluded for plane wave incidence. For perfect crystals, the A -matrix may be block diagonalised. This leads to a considerable increase in the computational efficiency of the model and yields important insights into the physics of convergent probes in perfect crystals—reciprocity in coherent imaging and the small aperture limit for coherent and incoherent contrast are considered. The numerical equivalence of the incoherent lattice contrast calculated in this Bloch wave method and the multislice method using mixed dynamic form factors will be demonstrated. Comparison between both these methods and the frozen phonon model, a prevalent multislice method for annular dark field simulation which has the theoretical advantage of handling double channelling, will be made.

Charge density analysis from complementary high energy synchrotron X-ray and electron diffraction data

To accurately measure the low order structure factors of α-Al 2O 3, two-dimensional convergent beam electron diffraction (CBED) data from parallel-sided platelets at various accelerating voltages, thickness and orientations have been matched using the Bloch-wave method. Middle and high angle extinction-free data has been obtained by the extrapolation of multi-wavelength synchrotron X-ray measurements to zero wavelength. The combination of high-energy synchrotron X-ray diffraction and CBED allows the extinction-free absolute-scale measurements and improves the reliability of the electron charge density maps in α-Al 2O 3.

A practical approach for STEM image simulation based on the FFT multislice method

It has been demonstrated that a high-angle annular dark-field (HAADF) STEM technique gives an image resolving atomic columns. Due to the diffusion of this technique and an improvement of its resolution, a practical procedure for image simulation becomes important for a quantitative interpretation of the HAADF image. In this report a new practical scheme for a STEM image simulation is developed based on the FFT multislice algorithm. Here, a HAADF intensity due to thermal diffuse scattering (TDS) is calculated from the absorptive potential corresponding to high-angle TDS and the wave function equivalent to the propagating probe within the sample. Contrary to the commonly used Bloch wave method, a coherent bright-field intensity and a coherent HAADF intensity are also obtained straightforwardly. The HAADF image contrast calculated for GaAs is not simply proportional to Z 2 as expected from the Rutherford scattering at high-angle, and the As/Ga contrast ratio depends on the specimen thickness. This suggests that the generation of the HAADF signal is appreciably affected by the coherent dynamical scattering. The developed procedure here will have a definitive advantage over the Bloch wave approach for simulating the HAADF images expected from a defect and interface or amorphous materials, and also the HAADF image obtained by using a Cs-corrected microscope. This is because the former requires a huge super cell, while the latter needs a large objective aperture including a large number of incident beam directions.