## Abstract

The averaged diffraction data alone cannot distinguish between models with different heterogeneous structures at length scales of about 2 nanometers, even when using high-resolution data. Although our approach to calculating diffraction intensities from the model differs from that of Roorda and Lewis, paracrystallinity in amorphous silicon is undeniably evident in the raw experimental fluctuation electron microscopy data.

The discrepancy in the reduced radial distribution function (RDF) plots arises because Roorda and Lewis (*1*) are probably calculating the RDF directly from our model atomic coordinates, which are periodic (*2*). Our experimentally constrained structural relaxation (ECSR) method is driven by diffraction information, which is not periodic. We match diffraction data to that computed from an isolated unit cell and ignore the periodicity of the model. There are pros and cons for each method for comparing to diffraction data. Our choice was motivated by a desire to preserve the valuable diffraction information that guides our structural relaxation procedure. For larger models, the two methods should converge to the same RDF.

We agree that higher-resolution diffraction data provides additional information and must therefore improve the structural accuracy. However, the RDF is a sample average and cannot tell us much about structural heterogeneity in amorphous samples, particularly in the diffraction-amorphous regime at structure-correlation length scales of about 2 nm. Diffraction data are collected over a large sample volume and so are homogeneously averaged. The data are then processed isotropically to produce a smoothed radial average, the RDF, which obviously cannot be correct for any individual atom in the disordered sample. A high-resolution RDF can inform us only about the quality of a model measured as a homogeneous average, and other topologically distinct models can also fit this averaged data. The central issue is to find the topologically correct model. The fact that a periodic CRN model matches the homogeneously averaged data does not mean that it is therefore the only structural model that can match that average. We have shown that both random and CRN models match the experimental RDF but produce flat calculated fluctuation electron microscopy (FEM) intensity variance, contrary to experiment. FEM diffraction intensity variance data provides additional information that shows that the homogeneous hypothesis (the CRN model) cannot be correct for a-Si.

To explore the claim that higher-resolution diffraction data will resolve the structure better at the 2-nm length scale, we computed a high-resolution diffraction pattern (angular wave vector *Q*_{max} = 40 Å^{−1}, using our nonperiodic diffraction method) from a CRN model, shown in projection in Fig. 1A. Using this model’s computed diffraction pattern as noise-free input data, along with the experimental FEM variance data, we applied the ECSR procedure to obtain the model shown in projection in Fig. 1B, which stubbornly exhibits paracrystalline order. The original CRN model is not recovered, even though it was the CRN’s high-resolution diffraction data that was used. The FEM data favor the paracrystallite model over the CRN. The reduced diffraction intensity plots for the two models (Fig. 1C) are essentially identical, overlapping almost perfectly, illustrating that averaged diffraction alone cannot tell the two models apart.

Consequently, Roorda and Lewis overreach with their claim that “High-resolution x-ray measurements support [the defective CRN model] fully.” The strongest statement that they are entitled to make about their measurements is that their experimental RDF is not inconsistent with their particular CRN model. The good agreement between averaged diffraction data alone does not prove that the CRN is the only possible model.

Roorda and Lewis estimate that the amount of paracrystalline material may be as low as 300 parts per million. This would correspond to half an atom in our 1728-atom models. Their estimate is too low. We have employed a stringent definition of a paracrystallite, which involves topological constraints out to the third-nearest neighbor to define a local cluster of 29 atoms (*3*, *4*). By one measure, we can say that this counts only as one paracrystalline atom, despite the entourage of 28 other atoms that help define it.

Roorda and Lewis remind us that models containing voids can also explain the FEM data (*5*). This translates into the extraordinary claim that voids provide a better explanation for the presence of crystalline 111, 220, and 311 intensity-variance peaks than do the topologically cubic-structure paracrystallites. In principle, void models are a legitimate part of the solution space that the ECSR method explores, and yet voids do not emerge as viable structures relative to paracrystallites. We have explored models that are seeded with voids at the start of ECSR processing, because there is the possibility that the reduced constraints at the internal void surfaces might encourage nucleation of paracrystallites. This nucleation did not happen in our ECSR runs. Nevertheless, despite the absence of paracrystallites, the final models matched both the diffraction and the variance data, although the Tersoff energy was high relative to typical paracrystallite models. The void-seeded models contained implausible medium-range correlations where {111}-type density modulations appeared in projection when the models were viewed at certain angles, but the {111} short-range structural correlations needed to construct crystallographically true {111} planes were absent. An example of this phenomenon, where medium-range order has little or no accompanying short-range order, is shown in (*6*). In any event, Roorda and Lewis state that small-angle x-ray scattering data argue against a mixed-phase structure. Shouldn’t this also rule out the void model? However, it is not so clear that this necessarily rules out the paracrystallite model, because the density fluctuations (silicon against silicon) are much more subtle.

Contrary to Roorda and Lewis’s claim that paracrystallites dissolve with increasing temperature, which is pointed out as an inconsistency with the physical expectation of crystallization, an earlier FEM study (*7*) on the a-Si to c-Si phase transition shows that there is a continuous evolution of an ordered Si phase as temperature increases. Diffraction data misleadingly show a “sudden” onset of crystallinity out of the diffraction-amorphous state much later than the actual onset of crystallization, as observed by FEM data. It cannot be overemphasized that homogeneous diffraction alone is insensitive to structural inhomogeneity smaller than about 2 nm in extent.

Roorda and Lewis declare that the FEM method “is at best qualitative.” Presumably, their view is that the RDF method is quantitative. A more valuable measure of these techniques is the information they provide about the sample. The RDF method, being an isotropic average of a two-body data set, tells us very little quantitatively about the sample—the average nearest neighbor distance, the average coordination number, and the mean density—and it determines these values precisely. Conversely, the FEM variance data are a complex four-body data set that is rich with details about spatial heterogeneity but is difficult to invert into a specific structure. Just as the mean tells us nothing about the variance, the variance tells us nothing about the mean. It is true that not all of the contributions to the FEM signal are understood fully—in particular, the decoherence effects that strongly suppress the diffraction speckle that FEM examines. It is not unreasonable to use hidden-parameter inferencing to refine the γ factor, which accounts phenomenologically for decoherence, along with the structural parameters. The γ factor is linearly independent of the other parameters. It should not be overlooked that the raw FEM data show strong evidence for {111}, {220} and {311} paracrystalline correlations before any numerical processing is done. The raw data alone are inconsistent with a CRN model.

Fluctuation EM, and the ECSR method for inverting the data, are still emerging techniques that can be improved. With modern-day scanning transmission electron microscopes, FEM is a straightforward technique to implement. The ECSR method is computationally intensive and can clearly be improved to work with bigger models. We maintain that the paracrystallite model for a-Si is broadly correct and supersedes the interrupted-CRN model favored by Roorda and Lewis, even for annealed material. More does need to be done to narrow further the set of paracrystalline macrostates that match the data. Although this is an old subject, there is still plenty of scope for further advances in our understanding of the structural details of amorphous silicon.