Higher-order Laue Zone Reflections Research Articles

On the origin of diffuse intensities in fcc electron diffraction patterns.

Interpreting diffuse intensities in electron diffraction patterns can be challenging in samples with high atomic-level complexity, as often is the case with multi-principal element alloys. For example, diffuse intensities in electron diffraction patterns from simple face-centred cubic (fcc) and related alloys have been attributed to short-range order1, medium-range order2 or a variety of different {111} planar defects, including thin twins3, thin hexagonal close-packed layers4, relrod spiking5 and incomplete ABC stacking6. Here we demonstrate that many of these diffuse intensities, including [Formula: see text]{422} and [Formula: see text]{311} in ⟨111⟩ and ⟨112⟩ selected area diffraction patterns, respectively, are due to reflections from higher-order Laue zones. We show similar features along many different zone axes in a wide range of simple fcc materials, including CdTe, pure Ni and pure Al. Using electron diffraction theory, we explain these intensities and show that our calculated intensities of projected higher-order Laue zone reflections as a function of deviation from their Bragg conditions match well with the observed intensities, proving that these intensities are universal in these fcc materials. Finally, we provide a framework for determining the nature and location of diffuse intensities that could indicate the presence of short-range order or medium-range order.

UnitCell Tools, a package to determine unit-cell parameters from a single electron diffraction pattern.

Electron diffraction techniques in transmission electron microscopy (TEM) have been successfully employed for determining the unit-cell parameters of crystal phases, albeit they exhibit a limited accuracy compared with X-ray or neutron diffraction, and they often involve a tedious measurement procedure. Here, a new package for determining unit-cell parameters from a single electron diffraction pattern has been developed. The essence of the package is to reconstruct a 3D reciprocal primitive cell from a single electron diffraction pattern containing both zero-order Laue zone and high-order Laue zone reflections. Subsequently, the primitive cell can be reduced to the Niggli cell which, in turn, can be converted into the unit cell. Using both simulated and experimental patterns, we detail the working procedure and address some effects of experimental conditions (diffraction distortions, misorientation of the zone axis and the use of high-index zone axis) on the robustness and accuracy of the software developed. The feasibility of unit-cell determination of the TiO2 nanorod using this package is also demonstrated. Should the parallel-beam, nano-beam and convergent-beam modes of the TEM be used flexibly, the software can determine unit-cell parameters of unknown-structure crystallites (typically >50 nm).

Validities of three multislice algorithms for quantitative low-energy transmission electron microscopy

Three different types of multislice algorithms, namely the conventional multislice (CMS) algorithm, the propagator-corrected multislice (PCMS) algorithm and the fully-corrected multislice (FCMS) algorithm, have been evaluated in comparison with respect to the accelerating voltages in transmission electron microscopy. Detailed numerical calculations have been performed to test their validities. The results show that the three algorithms are equivalent for accelerating voltage above 100kV. However, below 100 kV, the CMS algorithm will introduce significant errors, not only for higher-order Laue zone (HOLZ) reflections but also for zero-order Laue zone (ZOLZ) reflections. The differences between the PCMS and FCMS algorithms are negligible and mainly appear in HOLZ reflections. Nonetheless, when the accelerating voltage is further lowered to 20 kV or below, the PCMS algorithm will also yield results deviating from the FCMS results. The present study demonstrates that the propagation of the electron wave from one slice to the next slice is actually cross-correlated with the crystal potential in a complex manner, such that when the accelerating voltage is lowered to 10 kV, the accuracy of the algorithms is dependent of the scattering power of the specimen.

Improvement of the precision of lattice parameter determination by nano-beam electron diffraction

A highly precise determination of lattice parameters using higher-order Laue zone (HOLZ) reflections observed in nano-beam electron diffraction is presented. The introduction of more than 40 HOLZ reflections, whose positions are corrected by considering the aberration of the electron optics and are determined with an accuracy of 0.04 nm⁻¹, allows us to achieve a remarkable high precision of a 0.02% error, which is four times higher than the precision without HOLZ reflections.

Effects of Higher-Order Laue Zone Reflections on Atomic Resolution TEM Images of Single Layer and Bilayer Graphene

Extended abstract of a paper presented at Microscopy and Microanalysis 2010 in Portland, Oregon, USA, August 1 – August 5, 2010.

Enantiomorph identification in organic crystals by electron diffraction

The convergent-beam electron diffraction (CBED) method we proposed recently for enantiomorph identification has successfully been applied to some amino acid crystals such as glutamic acid and threonin. Enantiomorph identification (either left-handed or right-handed form) can readily be made within the framework of the proposed method by noting the asymmetric intensity distribution of Bijvoet pairs of reflections in the CBED pattern taken along an appropriate zone-axis orientation. Although the proposed method usually requires only a single CBED pattern, some effort to eliminate the ambiguity of 180°-rotation of the CBED pattern about the incident beam is needed for enantiomorph identification for these organic crystals because of the lack of HOLZ (higher-order Laue zone) reflection disks.

Structure refinement of a decagonal quasicrsytal of Al72Ni20Co8 by convergent-beam electron diffraction

Abstract Structure refinement of a water quenched alloy of Al72Ni20Co8, which is known as a highly ordered decagonal quasicrystal, has been carried out by quantitative comparison of experimental and simulated intensities of convergent-beam electron diffraction (CBED) patterns. In the present study, higher-order Laue zone reflections, whose intensities are senstively changed with atom positions, have been used for the refinement of atom position parameters in the quasiperiodic directions. The refined parameters have been compared with the results obtained by X-ray diffraction.

Dynamical simulations of zone axis electron channelling patterns of cubic silicon carbide

We have observed and simulated energy-dependent intensity distributions in electron channelling patterns (ECP) of cubic silicon carbide (3C SiC) which were recorded close to the (1 1 1) zone axis. The kinetic energies used were in the range from 4 to 8 keV , covering the low-energy region of the ECP technique. We explain the observed patterns by dynamical many beam simulations using a bloch wave approach for the diffraction of the incoming beam and the forward–backward-approximation for the backscattering of the electrons. The dynamical simulations reproduce the experimental patterns very well. It is found that higher-order Laue zone reflections are responsible for the strong energy sensitivity of the intensity distributions.

Simulation of electron channelling patterns for in situ polytype analysis of silicon carbide films

For the controlled growth of hexagonal or cubic silicon carbide (SiC) polytypes, an in situ surface analysis technique is needed which is sensitive to the crystal structure of thin films whose thicknesses are in the range of a few nanometers. Diffraction effects of electrons having kinetic energies of a few keV serve this purpose very well. X-ray photoelectron diffraction patterns and electron channelling patterns (ECP) have been proven to be useful techniques for polytype differentiation. We show that the polytype specific effects in zone axis ECP of SiC polytypes can be simulated using a Bloch wave approach. Dynamical diffraction leads to characteristic patterns which show a variety of energy dependent structures. The specific zone axis patterns are caused by many-beam effects of higher order Laue zone reflections which are strongly energy dependent. We show results for 3C, 4H and 6H SiC.

Nanometer Crystal Structure Analysis by EF-CBED and EF-Microscopy

Abstract Crystal structure refinement by the convergent-beam electron diffraction (CBED) method has the following advantages: (i) Nanometer-size crystal structure refinement: CBED patterns can be obtained from specimen areas of a few nanometer in diameter, (ii) Dynamical diffraction effect: the CBED intensities contain phase information of crystal structure factors through the strong dynamical effect. (iii) Site-selective analysis: the use of Bloch states formed by incident electrons allows structure determination weighted for specific atom sites. Energy-filtering (EF) to revome inelastically scattered background is now getting necessary for such quantitative analysis of CBED patterns. Especially for the determination of atom positions and Debye-Waller factors, the use of energy-filtered higher-order Laue zone (HOLZ) CBED patterns is essential because small displacements of atoms can be sensitively detected using HOLZ reflections with large reciprocal vectors. For this purpose, we developed a new fl-filter transmission electron microscope (JEM-2010FEF) with a high acceptance angle [2] and a new analysis program “mbfit” to refine structural parameters using many-beam Bloch-wave calculations and least-square fitting [1].

A dynamical Bragg equation for high-order Laue zone reflections

A dynamically corrected Bragg equation for high-order Laue zone (HOLZ) reflections is derived directly from the Bloch wave formalism instead of the geometric argument used to deduce the kinematical Bragg condition. It differs from the kinematical Bragg equation by replacing the plane wave vector in the kinematical equation with the Bloch wave vectors. This dynamical equation reduces to the kinematical equation when the crystal potential is zero. It also demonstrates the occurrence of dynamical shifts for the HOLZ reflections but their absence for the zero-order Laue zone (ZOLZ) reflections in the symmetrical Laue case.

Refinement of crystal structural parameters using two-dimensional energy-filtered CBED patterns.

A new method to refine crystal structural parameters using convergent-beam electron diffraction (CBED), which is applicable to nanometre-size crystal structure analysis, is proposed. This method is based on the fitting between theoretical calculations and experimental intensities of energy-filtered two-dimensional CBED patterns containing higher-order Laue-zone (HOLZ) reflections. The use of HOLZ reflections is essential for the method because small displacements of atoms can be sensitively detected using HOLZ reflections with large reciprocal vectors. For this purpose, a new Omega-filter transmission microscope (JEM-2010FEF), which can take energy-filtered CBED patterns up to a high angle with a small distortion, and a new analysis program to refine structural parameters, which is based on many-beam Bloch-wave calculations and nonlinear least-squares fitting, have been developed. As a test example, a positional parameter and isotropic and anisotropic Debye-Waller factors of CdS have been refined. Two-dimensional CBED patterns calculated with the refined parameters show very good agreement with the experimental ones, and the refined values of the parameters also agree well with the result of a single-crystal X-ray diffraction experiment. Important problems of the analysis procedure are discussed item by item.

Multislice Method for Large Beam Tilt with Application to HOLZ Effects in Triclinic and Monoclinic Crystals

Three existing multislice formulations (MS) that are claimed to be applicable for large beam tilt are introduced in a unified form and the validity of these formulations for large beam tilt is evaluated. It is shown that one formula, which is called MSSBT (the MS for small beam tilt), is valid only for tilt angles less than 6°, but the second one, which is called MSLBT (the MS for large beam tilt), can be used for tilt angles as large as 20°, while the third one yields results very close to the MSLBT results for tilt angles up to 12°. Simulations also show that the accuracy of the MSLBT is independent of the scattering power of the atoms. The reasons for the inaccuracy of these multislice formulations for beam tilt (MSBT) and the differences between the three MSBT formulations are discussed based on the complete Schrödinger equation, which includes back-scattering effects. For calculating the higher-order Laue-zone (HOLZ) effects from triclinic and monoclinic crystals, it is pointed out that the slices have to be cut parallel to the ab plane of the crystal and MSBT formulations should be used. The computational formulations for potentials of the crystal slices and the propagator are derived based on the transformations between the crystallographic coordinate system and the working coordinate system. Calculations of the HOLZ reflections for the monoclinic crystal Na2Ti3O7 [001] are carried out. It is shown that for triclinic and monoclinic (when the beam direction is along the c axis) crystals HOLZ effects dynamically influence not only the intensity but also the symmetry of zero-order Laue-zone (ZOLZ) diffraction patterns. Hence, in that case, the projection approximation for simulating the HRTEM images may not be used.

The Determination of Arsenic Atom Concentration in Arsenic-Doped Silicon by Using HOLZ Analysis

Higher-order Laue zone (HOLZ) disk patterns obtained from arsenic-doped and undoped silicon are investigated under the exact [011] zone axis orientation at 100 kV. For this orientation, the fine HOLZ structure consists mainly of two bright lines produced from the intersections of a HOLZ reflection with two branches associated with atomic and tunneling sites of the zeroth-layer dispersion surfaces. The arsenic atom concentration and its strain contribution can be visibly detected by comparison of the two HOLZ lines on doped silicon with those on the undoped one. It is revealed that the position for these lines is not sensitive to the crystal thickness, Debye-Waller factors, and absorption potential but to the impurity concentration and accelerating voltage. The simple fitting between experiment and dynamical simulation permits a reasonable estimation of the arsenic atom concentration.

Higher-order Laue zone contrast from arsenic-doped silicon

Higher-order Laue zone (HOLZ) disc patterns obtained from As-doped and undoped Si have been examined under the exact 011 zone axis orientation at 100 kV. For this orientation, two lines in the HOLZ disc are produced from the intersections of HOLZ reflections, with two branches corresponding to atomic and tunnelling sites of the zeroth-layer dispersion surface. The As atom concentration and its strain contribution can be visibly detected by comparison of the two HOLZ line positions. A simple fitting between experiment and simulation permits a reasonable estimate of the As atom concentration.

Can the Multislice Method Be Used To Calculate HOLZ Reflections in High-Energy Electron Diffraction and Imaging?

The validity of the use of the conventional multislice (MS) method for the calculation of higher-order Laue zone (HOLZ) reflections is tested by detailed calculations as well as a theoretical analysis. It is shown that if sufficiently thin slices are employed and the Debye-Waller (DW) factors for atoms are included the MS method can calculate HOLZ effects correctly up to the exact solution of the modified Schrodinger equation for high-energy electron diffraction. It is pointed out that the MS method is accurate up to second order in the slice thickness for zero-order Laue zone (ZOLZ) reflections but only up to first order for HOLZ reflections. As a consequence, the accurate calculation of HOLZ reflections requires a much smaller slice thickness. Using MS procedures in the standard way can then lead to accurate ZOLZ intensities but inaccurate HOLZ intensities. Without the introduction of DW factors for atoms, the atomic cores act as points of singularity for the MS formula and may cause severe errors for both the calculated HOLZ and ZOLZ reflections. It is also pointed out that if no so-called lower order HOLZ reflections are involved, one can simply use the projection approximation in HRTEM image simulations.

Dynamical diffraction of double-inelastically scattered electrons

Abstract Dynamical theories are developed to calculate the diffraction intensities of double-inelastically scattered high-energy electrons (i.e. the electrons that have been inelastically scattered twice as the result of exciting two distinct crystal states) in crystals within the framework of quantum mechanics. These theories are needed to quantify the data of thermal diffusely scattered electron energy loss spectroscopy. The Bloch wave and Green's function approaches proposed here take into account the full dynamical diffraction effects of the electrons before and after each inelastic event. The Bloch wave theory gives a full three-dimensional description of double-inelastic scattering in crystals including higher-order Laue zone reflections, but it may not be convenient to calculate the intensity of high-angle thermal diffuse scattering of electrons because of the difficulty of including all of the possible multiphonon excitations. Green's function theory, as an alternative approach, is most suitable f...

Dynamical diffraction effect on HOLZ-pattern geometry in Si-Ge alloys and determination of local lattice parameter

Higher-order Laue zone (HOLZ) patterns are observed in convergent-beam electron diffraction (CBED) patterns of Si-Ge alloys and the dynamical effect on HOLZ lines is discussed in relation to the kinematical approximation to simulate the HOLZ lines. Double lines often appear from one HOLZ reflection due to the excitation of the first and second branches of the zeroth-layer dispersion surface. The magnitude of the systematic line shift from the line positions predicted by the kinematical simulation changes sensitively with specimen thickness and alloy composition depending on which branch is dominantly excited. The kinematical simulation is applicable to accurate determination of lattice parameters with a proper correction δ E: a difference between the effective electron energy and the actual microscope operating voltage. Usefulness of the kinematical simulation is demonstrated in the investigation of the segregation in a Si-Ge alloy.

Effects of Higher-Order Laue Zone reflections on structure images

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

Formation of double-diffracted higher-order Laue zone lines

In crystals with a glide plane perpendicular to the incident beam, higher-order Laue zone (HOLZ) lines formed by double diffraction (DD)Mayappear as fine bright lines in the forbidden zero-order Laue zone discs of a convergent-beam electron diffraction pattern. In this paper the formation of DD HOLZ lines has been studied systematically. A pseudo-kinematical analysis has been carried out, enabling the positions and directions of DD HOLZ lines to be predicted. Simulations based on this analysis agree well with experimental results. Using the pseudo-kinematical formulation it can be demonstrated that extinction exists between overlapping conjugately paired DD HOLZ reflections as a result of opposite signs of the combined structure factors and approximately equal contributions of excitation error terms for the two reflections. In the dynamical analysis, a method to display branches of the dispersion surface has been devised and extensively used to investigate the calculated results of the Bloch wave dynamical theory related to DD HOLZ lines. The principal areas on the dispersion surface which contribute to the DD HOLZ line intensity have been located. The apparent extinction line between two conjugately paired DD HOLZ lines is due to incomplete destructive interference between two waves with equal amplitudes, opposite initial phases and slightly different eigenvalues (wave-vectors); this results in an approximate cancellation. By contrast, the GjØnnes-Moodie dynamical extinction condition corresponds to a situation where the degeneracy of eigenvalues occurs, resulting in completely destructive interference between waves and leading to exact cancellation.