Eric Mandell

Making sense of nanocrystal lattice fringes

The orientation dependence of thin-crystal lattice fringes can be gracefully quantified using fringe-visibility maps, a direct-space analog of Kikuchi maps [Nishikawa and Kikuchi, Nature (London) 121, 1019 (1928)]. As in navigation of reciprocal space with the aid of Kikuchi lines, fringe-visibility maps facilitate acquisition of crystallographic information from lattice images. In particular, these maps can help researchers to determine the three-dimensional lattice of individual nanocrystals, to “fringe-fingerprint” collections of randomly oriented particles, and to measure local specimen thickness with only a modest tilt. Since the number of fringes in an image increases with maximum spatial-frequency squared, these strategies (with help from more precise goniometers) will be more useful as aberration correction moves resolutions into the subangstrom range.

Quasi-adiabatic clocking of quantum-dot cellular automata

We present a theoretical study of quasi-adiabatic clocking of quantum-dot cellular automata (QCA). Quasi-adiabatic clocking refers to periodical modulation of interdot potential barriers in order to keep the cells of a QCA device near their ground state throughout the entire switching process. The barrier modulation has been studied through the use of a trapezoidal-shaped, periodic, time-dependent, electric field. The time-dependent electric field has been calculated for arrays of linear charged rods. A continuous traveling maximum in the electric field represents the flow of information from one zone to the next. For a QCA device where the zones are set up, such that the flow of information is linear, a line of electrostatically charged rods can quasi-adiabatically clock the system.

Fringe-Covariance "Fingerprinting" of Nanoparticle Lattice Images

Individual diffraction patterns only provide 2nd-moment (e.g. atom-atom correlation) data on specimens, while HRTEM and z-contrast images provide information on higher order correlations as well. Variable coherence-width microscopy [1], and high-tilt lattice-parameter determination [2], involve ways to quantitatively characterize higher order (e.g. pair-pair) correlation by examining data from more than one image or diffraction pattern. Here we discuss a simple strategy for quantifying information of this sort found in a single lattice-fringe image of multiple nano-particles. The theory we discuss addresses the kinds of correlations expected if the nanoparticles are randomly oriented. On the experimental end, the strategy is simple. Find all nanoparticles showing cross-fringes in the image, and draw one or more pairs of xs on a plot of interspot angle versus lattice spacing. The angles and spacings might be measured directly, or perhaps better from a small power spectrum superposed on each particle of interest. Each cross-fringe pair will result in marks at two spacings (one for each spot in the power spectrum) and at a common interspot angle value. Examples of such patterns, for 10 nm particles in a WC1-x plasma enhanced CVD film (Fig. 1) and TiO2 aerosol catalyst particles wet deposited on a holey carbon film (Fig. 2). The data for the foregoing figures was obtained manually. Nonetheless, the figures offer a scatter diagram characteristic of both the primary zones expected for randomly-oriented particles of each crystal type, and the range of spacing errors resulting from both foreshortened projection (more significant for smaller particles) and measurement error. Automated analysis may become practical in the future, for example by tiling the image on a size scale characteristic of the grain size of interest, and then using power spectrum peak analysis to determine what to plot where. What if we now wish to generate a theoretical fingerprint for a given crystal type? To this end, we consider maps of angular covariances, i.e. the average product of power spectrum intensity at two reciprocal spacings, separated in the three dimensional reciprocal-lattice by an interspot angle of ∆α. Suitable normalization, with means and standard deviations, allows one to thereby map a kind of angular correlation coefficient across the space of scattering probabilities. A plot of such correlations in the (hk0) plane of a simple cubic lattice is shown in Fig. 3. The maps for comparison with a given set of data will depend on both crystallite size (e.g. fringe foreshortening effects are more prevalent in smaller crystals) and the microscopes instrument response function (e.g. even single zone axis patterns will show more fringe pairs with better contrast transfer). Links to angular covariance maps for some commonly encountered structures will made available on the web [3].

Measuring Local Thickness Through Small-Tilt Fringe Visibility

Specimen thickness measurements are often limited to analyzing one region at a time, and by the size of the electron probe. With increased availability of lattice fringe data in phase and z contrast images, information on how fringes change with tilt is also more accessible. We discuss how such data can provide thickness information on specimen regions only nanometers on a side, provided they are thin enough for lattice imaging. Using micrographs 1/3 micron across, many regions can be analyzed with only a few images.

Darkfield Brightfield and Energy-Filtered Nanotube Image Profiles

Anyone who has been asked to examine carbon nanotubes in the TEM knows that cylindrical symmetry can be a wonderful thing. For example, if you would like to know how many graphene or BN sheets make up a given tube, or whether or not the tube has internal terminations, one image may do the job. In this paper, we discuss projected thickness and diffraction functions that provide a baseline for examining a variety of intensity profiles across nanotube images. These likewise assume that to first order the tube is cylindrically symmetric, the beam encounters the tube perpendicular to its symmetry axis, and that the contrast mechanisms are simple. One can always explore deviations from these assumptions after the fact.