Entropic Comparison of Atomic-Resolution Electron Tomography of Crystals and Amorphous Materials.

Electron tomography bears promise for widespread determination of the three-dimensional arrangement of atoms in solids. However, it remains unclear whether methods successful for crystals are optimal for amorphous solids. Here, we explore the relative difficulty encountered in atomic-resolution tomography of crystalline and amorphous nanoparticles. We define an informational entropy to reveal the inherent importance of low-entropy zone-axis projections in the reconstruction of crystals. In turn, we propose considerations for optimal sampling for tomography of ordered and disordered materials.

Fluctuation microscopy analysis of amorphous silicon models.

Using computer-generated models we discuss the use of fluctuation electron microscopy (FEM) to identify the structure of amorphous silicon. We show that a combination of variable resolution FEM to measure the correlation length, with correlograph analysis to obtain the structural motif, can pin down structural correlations. We introduce the method of correlograph variance as a promising means of independently measuring the volume fraction of a paracrystalline composite. From comparisons with published data, we affirm that only a composite material of paracrystalline and continuous random network that is substantially paracrystalline could explain the existing experimental data, and point the way to more precise measurements on amorphous semiconductors. The results are of general interest for other classes of disordered materials.

Flexibility mechanisms in ideal zeolite frameworks.

Zeolites are microporous crystalline aluminosilicate materials whose atomic structures can be usefully modelled in purely mechanical terms as stress-free periodic trusses constructed from rigid corner-connected SiO4 and AlO4 tetrahedra. When modelled this way, all of the known synthesized zeolite frameworks exhibit a range of densities, known as the flexibility window, over which they satisfy the framework mechanical constraints. Within the flexibility window internal stresses are accommodated by force-free coordinated rotations of the tetrahedra about their apices (oxygen atoms). We use rigidity theory to explore the folding mechanisms within the flexibility window, and derive an expression for the configurational entropic density throughout the flexibility window. By comparison with the structures of pure silica zeolite materials, we conclude that configurational entropy associated with the flexibility modes is not a dominant thermodynamic term in most bulk zeolite crystals. Nevertheless, the presence of a flexibility window in an idealized hypothetical tetrahedral framework may be thermodynamically important at the nucleation stage of zeolite formation, suggesting that flexibility is a strong indicator that the topology is realizable as a zeolite. Only a small fraction of the vast number of hypothetical zeolites that are known exhibit flexibility. The absence of a flexibility window may explain why so few hypothetical frameworks are realized in nature.

The local structure of amorphous silicon.

It is widely believed that the continuous random network (CRN) model represents the structural topology of amorphous silicon. The key evidence is that the model can reproduce well experimental reduced density functions (RDFs) obtained by diffraction. By using a combination of electron diffraction and fluctuation electron microscopy (FEM) variance data as experimental constraints in a structural relaxation procedure, we show that the CRN is not unique in matching the experimental RDF. We find that inhomogeneous paracrystalline structures containing local cubic ordering at the 10 to 20 angstrom length scale are also fully consistent with the RDF data. Crucially, they also matched the FEM variance data, unlike the CRN model. The paracrystalline model has implications for understanding phase transformation processes in various materials that extend beyond amorphous silicon.

Examination of a polycrystalline thin-film model to explore the relation between probe size and structural correlation length in fluctuation electron microscopy.

We examine simulated electron microdiffraction patterns from models of thin polycrystalline silicon. The models are made by a Voronoi tessellation of random points in a box. The Voronoi domains are randomly selected to contain either a randomly-oriented cubic crystalline grain or a region of continuous random network material. The microdiffraction simulations from coherent probes of different widths are computed at the ideal kinematical limit, ignoring inelastic and multiple scattering. By examining the normalized intensity variance that is obtained in fluctuation electron microscopy experiments, we confirm that intensity fluctuations increase monotonically with the percentage of crystalline grains in the material. However, anomalously high variance is observed for models that have 100% crystalline grains with no imperfections. We confirm that the reduced normalized variance, V(k,R) - 1, that is associated with four-body correlations at scattering vector k, varies inversely with specimen thickness. Further, for probe sizes R larger than the mean grain size, we confirm that the reduced normalized variance obeys the predicted form given by Gibson et al. [Ultramicroscopy, 83, 169-178 (2000)] for the kinematical coherent scattering limit.

A surprise in the first Born approximation for electron scattering.

A standard textbook derivation for the scattering of electrons by a weak potential under the first Born approximation suggests that the far-field scattered wave should be in phase with the incident wave. However, it is well known that waves scattered from a weak phase object should be phase-shifted by π/2 relative to the incident wave. A disturbing consequence of this missing phase is that, according to the Optical Theorem, the total scattering cross section would be zero in the first Born approximation. We resolve this mystery pedagogically by showing that the first Born approximation fails to conserve electrons even to first order. Modifying the derivation to conserve electrons introduces the correct phase without changing the scattering amplitude. We also show that the far-field expansion for the scattered waves used in many texts is inappropriate for computing an exit wave from a sample, and that the near-field expansion also give the appropriately phase-shifted result.

Density of mechanisms within the flexibility window of zeolites.

By treating idealized zeolite frameworks as periodic mechanical trusses, we show that the number of flexible folding mechanisms in zeolite frameworks is strongly peaked at the minimum density end of their flexibility window. 25 of the 197 known zeolite frameworks exhibit an extensive flexibility, where the number of unique mechanisms increases linearly with the volume when long wavelength mechanisms are included. Extensively flexible frameworks therefore have a maximum in configurational entropy, as large crystals, at their lowest density. Most real zeolites do not exhibit extensive flexibility, suggesting that surface and edge mechanisms are important, likely during the nucleation and growth stage. The prevalence of flexibility in real zeolites suggests that, in addition to low framework energy, it is an important criterion when searching large databases of hypothetical zeolites for potentially useful realizable structures.

Substantial crystalline topology in amorphous silicon.

Using electron correlograph analysis we show that coherent nanodiffraction patterns from sputtered amorphous silicon indicate that there is more local crystallinity in unannealed amorphous silicon than was previously suspected. By comparing with simulations for various models we show that within a typical unannealed amorphous silicon film a substantial volume fraction (>50%) is topologically crystalline with correlation lengths up to 2 nm. Electron correlograph analysis is a variant of the fluctuation electron microscopy technique and its sensitivity to local crystalline ordering is derived from its sensitivity to four-body correlations.

Flexibility of ideal zeolite frameworks.

We explore the flexibility windows of the 194 presently-known zeolite frameworks. The flexibility window represents a range of densities within which an ideal zeolite framework is stress-free. Here, we consider the ideal zeolite to be an assembly of rigid corner-sharing perfect tetrahedra. The corner linkages between tetrahedra are hard-sphere oxygen atoms, which are presumed to act as freely-rotating, force-free, spherical joints. All other inter-tetrahedral forces, such as coulomb interactions, are ignored. Thus, the flexibility window represents the null-space of the kinematic matrix that governs the allowable internal motions of the ideal zeolite framework. We show that almost all of the known aluminosilicate or aluminophosphate zeolites exhibit a flexibility window. Consequently, the presence of flexibility in a hypothetical framework topology promises to be a valuable indicator of synthetic feasibility. We describe computational methods for exploring the flexibility window, and discuss some of the exceptions to this flexibility rule.

Medium-range order in molecular materials: fluctuation electron microscopy for detecting fullerenes in disordered carbons.

During a fluctuation electron microscopy (FEM) study of disordered carbons, we found that samples containing C(60) exhibit a normalized variance peak at 7.1 nm(-1) that appears to be a unique indicator of tight curvature in layered materials. This peak is associated with the characteristic in-plane carbon-carbon bond distance of approximately 0.14 nm in graphene. Diffraction from this spacing is normally forbidden in planar graphene (and graphite), but becomes allowed when the layer structure is interrupted. Such interruptions arise at the edges of graphite fragments and also when 5-rings are incorporated into a layer. We show that the curvature induced by a high density of 5-rings, such as that in C(60), can dominate the variance peak at 7.1 nm(-1). FEM simulations reveal that the variance peak at approximately 7.1 nm(-1), which we label F(1), is one of several fullerene-signature peaks, with others occurring at Q values of 10.6 nm(-1) (F(2)) and 12.4 nm(-1) (F(3)). We conclude that FEM is a sensitive method for detecting dilute quantities of highly curved pentagon-rich fullerenes, such as C(60), when dispersed within disordered graphitic carbon.

Fluctuation X-ray microscopy: a novel approach for the structural study of disordered materials.

Measuring medium-range order is a challenging and important problem in the structural study of disordered materials. We have developed a new technique, fluctuation x-ray microscopy, that offers quantitative insight into medium-range correlations in disordered materials at nanometre and larger length scales. In this technique, which requires a spatially coherent x-ray beam, a series of speckle patterns are measured at a large number of locations in a sample using various illumination sizes. Examination of the speckle variance as a function of the illumination spot size allows the structural correlation length to be measured. To demonstrate this technique we have studied polystyrene latex spheres, which serve as a model for a dense random-packed glass, and for the first time have measured the correlation length in a disordered system by fluctuation X-ray microscopy. We discuss data analysis and procedures to correct for shot noise and detector noise. This approach could be used to explore medium-range order and subtle spatial structural changes in a wide range of disordered materials, from soft matter to nanowire arrays, semiconductor quantum dot arrays and magnetic materials.

When structural noise is the signal: speckle statistics in fluctuation electron microscopy.

In a recent paper, Jungk et al. [Ultramicroscopy 104 (2005) 206-219] conclude that standard fluctuation electron microscopy speckle analysis, which examines the normalized variance of scattered intensity as a function of scattering angle, does not provide more information about disordered materials than does classical diffraction. In this letter, I point out flaws in both their experiments and their modeling that led them to this erroneous conclusion. In their experiments, they use resolutions that are excessively high relative to the length scale of the medium-range ordering in their samples, and consequently the diffraction fluctuations they seek are very weak. In their modeling, they attempt to correct this by applying low-pass image processing filters in order to broaden the effective point-spread function (i.e. to lower the image resolution after the experiment). Such filters, when applied to image intensities, do not emulate properly the evolution of coherent speckle as the point-spread function becomes wider. I affirm that fluctuation microscopy does indeed provide more information about disordered materials than classical diffraction when the experiments and modeling are conducted appropriately.

The flexibility window in zeolites.

Today synthetic zeolites are the most important catalysts in petrochemical refineries because of their high internal surface areas and molecular-sieving properties. There have been considerable efforts to synthesize new zeolites with specific pore geometries, to add to the 167 available at present. Millions of hypothetical structures have been generated on the basis of energy minimization, and there is an ongoing search for criteria capable of predicting new zeolite structures. Here we show, by geometric simulation, that all realizable zeolite framework structures show a flexibility window over a range of densities. We conjecture that this flexibility window is a necessary structural feature that enables zeolite synthesis, and therefore provides a valuable selection criterion when evaluating hypothetical zeolite framework structures as potential synthetic targets. We show that it is a general feature that experimental densities of silica zeolites lie at the low-density edge of this window--as the pores are driven to their maximum volume by Coulomb inflation. This is in contrast to most solids, which have the highest density consistent with the local chemical and geometrical constraints.

Absence of an abrupt phase change from polycrystalline to amorphous in silicon with deposition temperature.

Using fluctuation electron microscopy, we have observed an increase in the mesoscopic spatial fluctuations in the diffracted intensity from vapor-deposited silicon thin films as a function of substrate temperature from the amorphous to polycrystalline regimes. We interpret this increase as an increase in paracrystalline medium-range order in the sample. A paracrystal consists of topologically crystalline grains in a disordered matrix; in this model the increase in ordering is caused by an increase in the grain size or density. Our observations are counter to the previous belief that the amorphous to polycrystalline transition is a discontinuous disorder-order phase transition.