We examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

To date, it is unclear whether chemical order (or disorder) is in any way connected to double exchange, electronic phase separation, or charge ordering (CO) in manganites. In this work, we carry out an atomic resolution study of the colossal magnetoresistant manganite La2−2xSr1+2xMn2O7 (LSMO). We combine aberration-corrected electron microscopy and spectroscopy with spectroscopic image simulations, to analyze cation ordering at the atomic scale in real space in a number of LSMO single crystals. We compare three different compositions within the phase diagram: a ferromagnetic metallic material (x=0.36), an insulating, antiferromagnetic charge ordered (AF-CO) compound (x=0.5), which also exhibits orbital ordering, and an additional AF sample (x=0.56). Detailed image simulations are essential to accurately quantify the degree of chemical ordering of these samples. We find a significant degree of long-range chemical ordering in all cases, which increases in the AF-CO range. However, the degree of ordering is never complete nor can it explain the strongly correlated underlying ordering phenomena. Our results show that chemical ordering over distinct crystallographic sites is not needed for electronic ordering phenomena to appear in manganites, and cannot by itself explain the complex electronic behavior of LSMO.

We present a theoretical framework for calculating probe-position-dependent electron energy-loss near-edge structure for the scanning transmission electron microscope by combining density functional theory with dynamical scattering theory. We show how simpler approaches to calculating near-edge structure fail to include the fundamental physics needed to understand the evolution of near-edge structure as a function of probe position and investigate the dependence of near-edge structure on probe size. It is within this framework that density functional theory should be presented, in order to ensure that variations of near-edge structure are truly due to local electronic structure and how much from the diffraction and focusing of the electron beam.

We show that aberration-corrected scanning transmission electron microscopy operating at low accelerating voltages is able to analyze, simultaneously and with single atom resolution and sensitivity, the local atomic configuration, chemical identities, and optical response at point defect sites in monolayer graphene. Sequential fast-scan annular dark-field (ADF) imaging provides direct visualization of point defect diffusion within the graphene lattice, with all atoms clearly resolved and identified via quantitative image analysis. Summing multiple ADF frames of stationary defects produce images with minimized statistical noise and reduced distortions of atomic positions. Electron energy-loss spectrum imaging of single atoms allows the delocalization of inelastic scattering to be quantified, and full quantum mechanical calculations are able to describe the delocalization effect with good accuracy. These capabilities open new opportunities to probe the defect structure, defect dynamics, and local optical properties in 2D materials with single atom sensitivity.

The relation between image resolution and information transfer is explored. It is shown that the existence of higher frequency transfer in the image is just a necessary but not sufficient condition for the achievement of higher resolution. Adopting a two-point resolution criterion, we suggest that a 10% contrast level between two features in an image should be used as a practical definition of resolution. In the context of scanning transmission electron microscopy, it is shown that the channeling effect does not have a direct connection with image resolution because sharp channeling peaks do not move with the scanning probe. Through a quantitative comparison between experimental image and simulation, a Fourier-space approach is proposed to estimate defocus and sample thickness. The effective atom size in Z-contrast imaging depends on the annular detector's inner angle. Therefore, an optimum angle exists for the highest resolution as a trade-off between reduced atom size and reduced signal with limited information transfer due to noise.

A real-space description of inelastic scattering in scanning transmission electron microscopy is derived with particular attention given to the implementation of the projected potential approximation. A hierarchy of approximations to expressions for inelastic images is presented. Emphasis is placed on the conditions that must hold in each case. The expressions that justify the most direct, visual interpretation of experimental data are also the most approximate. Therefore, caution must be exercised in selecting experimental parameters that validate the approximations needed for the analysis technique used. To make the most direct, visual interpretation of electron-energy-loss spectroscopic images from core-shell excitations requires detector improvements commensurate with those that aberration correction provides for the probe-forming lens. Such conditions can be relaxed when detailed simulations are performed as part of the analysis of experimental data.

In an article published in Microscopy and Microanalysis recently (Jia et al., 2004), it was claimed that aberration-corrected high resolution transmission electron microscopy (HRTEM) allows the quantitative measurement of oxygen concentrations in ceramic materials with atomic resolution. Similar claims have recently appeared elsewhere, based on images obtained through aberration correction (Jia et al., 2003; Jia & Urban, 2004) or very high voltages (Zhang et al., 2003). Seeing oxygen columns is a significant achievement of great importance (Spence, 2003) that will doubtlessly allow some exciting new science; however, other models could provide a better explanation for some of the experimental data than variations in the oxygen concentration. Quantification of the oxygen concentrations was attempted by comparing experimental images with simulations in which the fractional occupancy in individual oxygen columns was reduced. The results were interpreted as representing nonstoichiometry within the bulk and at grain boundaries. This is plausible because previous studies have shown that grain boundaries can be nonstoichiometric (Kim et al., 2001), and it is indeed possible that oxygen vacancies are present at boundaries or in the bulk. However, is this the only possible interpretation? We show that for the thicknesses considered a better match to the images is obtained using a simple model of surface damage in which atoms are removed from the surface, which would usually be interpreted as surface damage or local thickness variation (from ion milling, for example).