Exit wave reconstruction at atomic resolution

An iterative method for exit wave function reconstruction based on wave function propagation in free space is presented. The method, which has the potential for application to many forms of microscopy, has been tailored to work with a through focal series of images measured in a high-resolution transmission electron microscope. Practical difficulties for exit wave reconstruction which are pertinent in this experimental environment are the slight incoherence of the electron beam, sample drift and its effect upon the defocus step size that can be utilised, and the number of image measurements that need to be made. To gauge the effectiveness of the method it is applied to experimental data that has been analysed previously using a maximum likelihood formalism (the MAL method).     

Reconstruction of the projected crystal potential from a periodic high-resolution electron microscopy exit plane wave function

A method based on a simulated annealing algorithm is applied for the reconstruction of the projected crystal potential belonging to a periodic high-resolution electron microscopy exit plane wave function. Using simulated exit plane wave functions of GaAs at different specimen thicknesses, the convergence behaviour and the accuracy of the algorithm are investigated. It is demonstrated that the reconstruction is possible even under strongly non-linear scattering conditions at small specimen thicknesses. Further, the convergence of the algorithm to an ambiguous solution beyond a certain specimen thickness is discussed.     

Object wave reconstruction by phase-plate transmission electron microscopy

A method is described for the reconstruction of the amplitude and phase of the object exit wave function by phase-plate transmission electron microscopy. The proposed method can be considered as in-line holography and requires three images, taken with different phase shifts between undiffracted and diffracted electrons induced by a suitable phase-shifting device. The proposed method is applicable for arbitrary object exit wave functions and non-linear image formation. Verification of the method is performed for examples of a simulated crystalline object wave function and a wave function acquired with off-axis holography. The impact of noise on the reconstruction of the wave function is investigated.     

Projected potential profiles across interfaces obtained by reconstructing the exit face wave function from through focal series

An iterative method for reconstructing the exit face wave function from a through focal series of transmission electron microscopy image line profiles across an interface is presented. Apart from high-resolution images recorded with small changes in defocus, this method works also well for a large defocus range as used for Fresnel imaging. Using the phase-object approximation the projected electrostatic as well as the absorptive potential profiles across an interface are determined from this exit face wave function. A new experimental image alignment procedure was developed in order to align images with large relative defocus shift. The performance of this procedure is shown to be superior to other image alignment procedures existing in the literature. The reconstruction method is applied to both simulated and experimental images.     

The effect of mechanical vibration and drift on the reconstruction of exit waves from a through focus series of HREM images

Experimental HREM images can show a limited resolution as a result of mechanical vibration and drift. In this paper the effect of such mechanical vibrations on the accuracy of the through focus exit wave reconstruction method is investigated for different thicknesses of a test structure of La₃Ni₂B₂N₃. A through-focus series of HREM images for this structure is simulated for different kinds of mechanical vibration corresponding to an information limit g of about 7 nm⁻¹: (1) no mechanical vibration, (2) isotropic mechanical vibration, and (3) several anisotropic mechanical vibrations. From these through-focus series the reconstructed exit wave is calculated (Ultramicroscopy 64 (1996) 109). The above isotropic and anisotropic mechanical vibrations have a large effect on the reconstructed exit waves when compared with the reconstructed exit wave without mechanical vibration, i.e. the range of amplitudes and phases in a reconstructed exit wave decreases and the background intensity increases. The initial thickness and orientation can be obtained using a least-squares refinement procedure (Acta Crystallogr. A 54 (1998) 91) when there is no mechanical vibration present. In the case of isotropic or anisotropic vibration, the refined thickness and orientation are likely to give wrong results depending on the size of the vibrations and on the number of significant reflections (which is related to the size of the unit cell, the thickness and the misorientation).     

Resolution extension and exit wave reconstruction in complex HREM

Direct methods in real and reciprocal space are developed for structural reversion. The direct method in real space involves the use of a novel method to retrieve the phase in the image plane using transport of intensity equation/maximum entropy method (TIE/MEM) and exit wave reconstruction by self-consistent propagation. Since the exit wave is restored from the complex signal in the image planes, no image model between the exit wave and image is assumed. The structural information in the reconstructed exit wave is then further extended by a "complex" maximum entropy method as a direct method in reciprocal space to extrapolate the phase to higher frequencies.     

Ultrahigh resolution imaging of local structural distortions in intergrowth tungsten bronzes

Details of the local structure of a complex tungsten bronze, K(x)WO(3) have been determined using focal series exit wave reconstruction. Octahedral rotations in different structural regions of the same crystal have been directly measured from the exit wave phase and correlated with variations in cation occupancy determined from the exit wave modulus.     

A method to determine the local surface profile from reconstructed exit waves

Reconstructed exit waves are useful to quantify unknown structure parameters such as the position and composition of the atom columns at atomic scale. Existing techniques provide a complex wave in a flat plane which is close to the plane where the electrons leave the atom columns. However, due to local deviation in the flatness of the exit surface, there will be an offset between the plane of reconstruction and the actual exit of a specific atom column. Using the channelling theory, it has been shown that this defocus offset can in principle be determined atom column-by-atom column. As such, the surface roughness could be quantified at atomic scale. However, the outcome strongly depends on the initial plane of reconstruction especially in a crystalline structure. If this plane is further away from the true exit, the waves of the atom columns become delocalized and interfere mutually which strongly complicates the interpretation of the exit wave in terms of the local structure. In this paper, we will study the delocalization with defocus using the channelling theory in a more systematic way.     

Reconstruction of the projected electrostatic potential in high-resolution transmission electron microscopy including phenomenological absorption

The projected electrostatic potential is reconstructed from a high-resolution exit wave function through a maximum-likelihood refinement algorithm. The theory of an already existing algorithm [1] is extended to include the effects of phenomenological absorption. Various tests with a simulated exit wave function of YBa₂Cu₃O₇ in [1 0 0] orientation used as a source show that the reconstruction is successful, regardless of the strongly differing scattering power of atomic columns, even for the case of strong dynamical diffraction. Object thickness, the amount of absorption, and a residual defocus aberration of the wave function—parameters often unknown or difficult to measure in experiments—can be determined accurately with the aid of the refinement algorithm in a self-consistent way. For the next generation of instruments, with information limits of 0.05 nm and better, reconstruction accuracies of better than 2% can be expected, which is sufficient to measure and display the structural and chemical information with the aid of an accurate projected potential map.     

Imaging columns of the light elements carbon, nitrogen and oxygen with sub Angstrom resolution.

It is reported that lattice imaging with a 300 kV field emission microscope in combination with numerical reconstruction procedures can be used to reach an interpretable resolution of about 80 pm for the first time. A retrieval of the electron exit wave from focal series allows for the resolution of single atomic columns of the light elements carbon, nitrogen, and oxygen at a projected nearest neighbor spacing down to 85 pm. Lens aberrations are corrected on-line during the experiment and by hardware such that resulting image distortions are below 80 pm. Consequently, the imaging can be aberration-free to this extent. The resolution enhancement results from increased electrical and mechanical stability of the instrument coupled with a low spherical aberration coefficient of 0.595 + 0.005 mm.     

Retrieval of object information by inverse problems in electron diffraction

The imaging of crystal defects by high-resolution transmission electron microscopy or with the help of the electron diffraction contrast technique is well known and routinely used. However, a direct and phenomenological analysis of electron micrographs is mostly not possible, but requires the application of image simulation and matching techniques. The trial-and-error matching technique is the indirect solution to the direct scattering problem applied to analyse the nature of the object under investigation. Alternatively, inverse problems as direct solutions of electron scattering equations can be deduced using either an invertible linearized eigenvalue system or a discretized form of the diffraction equations. This analysis is based on the knowledge of the complex electron wave at the exit plane of an object reconstructed for the surrounding of single reflections by electron holography or other wave reconstruction techniques. In principle, it enables directly the retrieval of the local thickness and orientation of a sample as well as the refinement of potential coefficients or the determination of the atomic displacements, caused by a crystal lattice defect, relative to the atom positions of the perfect lattice. Considering especially the sample orientation as perturbation the solution is given by a generalized and regularized Moore–Penrose inverse, where the resulting numerical algorithms imply ill-posed inverse problems.     

Exit wave reconstruction using a conventional transmission electron microscope

Exit wave reconstruction of a focus series of Ge in [110] using the PAMMAL algorithm was performed on a conventional electron microscope. The simulated images using the reconstructed object wave match very well with those obtained experimentally. Amplitudes from the complex wave function were extracted by means of local Fourier transformation. Crystal thickness and tilt were determined locally by quantitative comparison of the reconstructed amplitudes with amplitudes from multislice calculations. Detailed analysis yields the quasicoherent imaging approach used in the PAMMAL algorithm to produce the largest error in the analysis. For the Ge crystal specimen parameters were quantified to spatial frequencies of 5 nm¹. In the case of an object producing strong diffracted beams, the reconstruction may fail because the quasicoherent approximation will not describe correctly the nonlinear image formation.     

Numerical analysis of impulse waves discharged at the exit of a model tunnel

Impulse waves are micro-pressure waves, which always occur at the tunnel exit when a high-speed train is moving inside a train tunnel. The air around the train nose is compressed and compression waves are induced. The impulse wave is discharged at the exit of a train tunnel when a compression wave propagates outside of the tunnel exit. Impulse waves are weak-strength pressure waves, which lead to noise and other environmental problems. In order to efficiently control the impulse wave at the exit of a train tunnel, numerical studies on investigating the generation and propagation of the impulse wave were carried out. A 2-D axisymmetric model tunnel was simulated at different operating conditions. Different Mach numbers of compression waves were varied to induce different magnitudes of impulse waves at the tunnel exit. In addition, compression waves with different pressure gradients were assumed at the tunnel entry to check their effects on the generation of impulse waves. In order to observe impulse waves at far field, five monitor points were installed behind the tunnel exit to record pressure histories as impulse waves moved through these locations. The detailed magnitudes and characteristics of impulse waves were obtained in the present studies.     

High precision measurements of atom column positions using model-based exit wave reconstruction

In this paper, it has been investigated how to measure atom column positions as accurately and precisely as possible using a focal series of images. In theory, it is expected that the precision would considerably improve using a maximum likelihood estimator based on the full series of focal images. As such, the theoretical lower bound on the variances of the unknown atom column positions can be attained. However, this approach is numerically demanding. Therefore, maximum likelihood estimation has been compared with the results obtained by fitting a model to a reconstructed exit wave rather than to the full series of focal images. Hence, a real space model-based exit wave reconstruction technique based on the channelling theory is introduced. Simulations show that the reconstructed complex exit wave contains the same amount of information concerning the atom column positions as the full series of focal images. Only for thin samples, which act as weak phase objects, this information can be retrieved from the phase of the reconstructed complex exit wave.     

Compositional analysis of mixed–cation-anion III–V semiconductor interfaces using phase retrieval high-resolution transmission electron microscopy

Employing exit‐plane wave function (EPWF) reconstruction in high‐resolution transmission electron microscopy (HRTEM), we have developed an approach to atomic scale compositional analysis of III‐V semiconductor interfaces, especially suitable for analyzing quaternary heterostructures with intermixing in both cation and anion sub‐lattices. Specifically, we use the focal‐series reconstruction technique, which retrieves the complex‐valued EPWF from a thru‐focus series of HRTEM images. A study of interfaces in Al_0.4Ga_0.6As–GaAs and In_0.25Ga_0.75Sb–InAs heterostructures using focal‐series reconstruction shows that change in chemical composition along individual atomic columns across an interface is discernible in the phase image of the reconstructed EPWF. To extract the interface composition profiles along the cation and anion sub‐lattices, quantitative analysis of the phase image is performed using factorial analysis of correspondence. This enabled independent quantification of changes in the In–Ga and As–Sb contents across ultra‐thin interfacial regions (approximately 0.6 nm wide) with true atomic resolution, in the In_0.25Ga_0.75Sb–InAs heterostructure. The validity of the method is demonstrated by analyzing simulated HRTEM images of an InAs–GaSb–InAs model structure with abrupt and graded interfaces. Our approach is general, permitting atomic‐level compositional analysis of heterostructures with two species per sub‐lattice, hitherto unfeasible with existing HRTEM methods.     

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions.

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.     

Electron channelling based crystallography

Electron channelling occurs when the incident electron beam is parallel to the atom columns of an object, such as a crystal or a particular crystal defect. Then, the electrons are trapped in the electrostatic potential of an atom column in which they scatter dynamically. This picture provides physical insight and explains why a one-to-one correspondence is maintained between the exit wave and the projected structure, even in case of strong dynamical scattering. Moreover, the theory is very useful to invert the dynamical scattering, that is, to derive the projected structure from the exit wave. Finally, it can be used to determine the composition of an atom column with single atom sensitivity or to explain dynamical electron diffraction effects. In this paper, an overview of the channelling theory will be given together with some recent applications.     

Performance limits of electron holography

Transmission electron microscopy is wave optics. The object exit wave contains the full object information. However, in the usual intensity images, recorded either in real space or in Fourier space, the phases are missing. In many applications at medium and at high resolution, electron holography has shown its unique ability of solving the “missing phase problem” and utilizing the recovered phase for complete interpretation of the object structure. The question is “What are the performance limits?” with respect to field of view, lateral resolution and signal resolution. In this article, the performance limits are derived and discussed.     

A computer program to illustrate macro topography on electron backscattering

This paper describes a computer program which enables the trajectories of an electron to be followed inside a solid sample and its pathway if it is successful to exit the sample's surface, and to be considered a backscattered electron (BSE). The simulation aids to understand the backscattering process and the effect of the sample that has other than an ideal flat surface topography. Several types of surface for the sample are simulated: a flat surface which can be tilted, a rough surface represented by a sine wave, and a circular surface to represent filamentous structures. The target material is entered as its empirical formula so that its mean atomic number may be calculated. It is covered with a layer of conductive coating, and it may be structured with vertical or horizontal layers of a second-phase material. Where appropriate, the position of the electron beam impacting on the target may be moved in relation to the underlying structures. The passage of the electron through the substrate is displayed on the computer monitor in high-resolution graphics. Information about those electrons which return to the sample surface and are backscattered is stored for later analysis. The paths of these BSEs are replotted to show the volume of the sample from which the image-forming data are derived. Further graphic output provides histograms of the energy distribution and of the BSEs exit direction.     

A simple channelling model for HREM contrast transfer under dynamical conditions

The application of electron channelling theory to dynamical exit wave calculations is briefly reviewed, and a comparison of channelling results with full dynamical calculations is presented. The channelling expression to the exit wave is combined with conventional imaging theory, and it is shown that a simple expression can be obtained for a dynamical contrast transfer function (D-CTF), which incorporates imaging aberrations and thickness-dependent dynamical scattering effects. The D-CTF can provide detailed insight into HREM images of a mixed cation oxide at thicknesses up to 200 Å, whereby an approximate correction for non-linear effects is utilized in the larger thickness regime.