title={Analysis of local strain in aluminium interconnects by energy filtered CBED},
title={Analysis of local strain in aluminum interconnects by energy filtered CBED},
author={Kr{\"a}mer, S and Mayer, J and Witt, C and Weickenmeier, A and R{\"u}hle, M},
journal={Ultramicroscopy},
volume={81},
@@ -422,3 +422,26 @@
title={Direct strain correlations at the single-atom level in three-dimensional core-shell interface structures},
journal={Nature Communications}
}
@article{4dstem_distortions,
title={Influence of distortions of recorded diffraction patterns on strain analysis by nano-beam electron diffraction},
author={Mahr, Christoph and M{\"u}ller-Caspary, Knut and Ritz, Robert and Simson, Martin and Grieb, Tim and Schowalter, Marco and Krause, Florian F and Lackmann, Anastasia and Soltau, Heike and Wittstock, Arne and others},
\def\blfootnote{\gdef\@thefnmark{}\@footnotetext}%Allow a blank unreferenced footnote to be placed
@@ -38,6 +46,7 @@
\affiliation{Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA}
\author{Jacob D. Hinkle\ \orcidlink{0000-0002-7751-1760}}
\affiliation{Computational Sciences \& Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA}
\affiliation{Currently at: NVIDIA}
\author{Jacob S. Spendelow\ \orcidlink{0000-0002-8111-7782}}
\affiliation{Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA}
\author{David A. Cullen\ \orcidlink{0000-0002-2593-7866}}
@@ -50,13 +59,13 @@
\date{\today}
\begin{abstract}
Lattice strain is one of the most fundamental tuning knobs available to materials designers, since bonding strength is directly modulated by strain. In most material systems, the lattice strain behavior present at the surface is different from the bulk crystal, with this behavior even more pronounced in nanoparticles. However, quantification of surface strain in nanoparticles remains a challenging endeavor, as the only technique that can reach the resolution needed to probe strain on a unit cell by unit cell basis, with the precision needed to track minute distortions is atomic resolution STEM or 4D-STEM. However, such techniques proposed till date can only visualize a single, or a few nanoparticles, as distortion quantification needs the nanoparticles to be oriented on-axis. Here we demonstrate 4D-STEM based quantification techniques that can \emph{simultaneously} quantify strain across hundreds of nanoparticles, even if none of those particles are on a low-index crystallographic axis. Furthermore, we show that combining the diffraction information from all the nanoparticle datasets can be used to generate diffraction space datasets that can track the evolution of strain across individual lattice planes.
Lattice strain is one of the most fundamental tuning knobs available to materials designers, since bonding strength is directly modulated by strain. In most material systems, the lattice strain behavior present at the surface is different from the bulk crystal, with this behavior even more pronounced in nanoparticles. However, quantification of surface strain in nanoparticles remains a challenging endeavor, as the only technique that can reach the resolution needed to probe strain on a unit cell by unit cell basis, with the precision needed to track minute distortions is transmission electron microscopy. However, such techniques proposed till date can only visualize a single, or a few nanoparticles. This is because strain quantification needs the nanoparticles to be oriented on-axis. Here we demonstrate 4D-STEM based quantification techniques that can \emph{simultaneously} quantify strain across hundreds of nanoparticles, even if none of those particles are on a low-index crystallographic zone axis. Furthermore, we show that combining the diffraction information from all the nanoparticle datasets can be used to generate diffraction space datasets that can track the evolution of strain across individual lattice planes.
\end{abstract}
\maketitle
\section{\label{sec:intro}Introduction}
\blfootnote{\textsf{This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (\href{http://energy.gov/downloads/doe-public-access-plan}{http://energy.gov/downloads/doe-public-access-plan})}}
\blfootnote{\textsf{This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (\url{http://energy.gov/downloads/doe-public-access-plan})}}
Over the past two centuries, anthropogenic activities have increased greenhouse gas (GHG) concentrations in the Earth's atmosphere as a byproduct of fossil fuel combustion \cite{ghg, humidity_temp}. The adverse effects of increasing GHGs are well studied and are already starting to be felt globally\cite{economic_climate_change}. As a result, alternative energy sources are being looked at, and of the multitude of candidates - hydrogen fuel cells have shown enormous promise for clean, compact energy generation systems\cite{hydrogen_fuel_cells}. In a fuel cell, hydrogen undergoes a redox reaction with oxygen with water as the byproduct. However, the oxygen reduction reaction (ORR) component of this redox reaction needs to be catalyzed \cite{strain_np_surface,HP_Pt_ORR,PEMFC_review}. Among the commonly used catalysis systems, platinum group metals (PGMs) are the most widely used system commercially. The problem with PGM catalysts is that such elements have low crustal abundance and are finite resources\cite{pt_abundance}. Thus, there has a significant research effort to decrease the PGM content of catalyst materials. Among such systems, platinum-cobalt alloys offer performance close to pure PGM metal catalysts while reducing the PGM loading \cite{core_shell_ordered_np,ultralow_ptco}. Such systems have been observed to demonstrate high mass activity that is maintained for over 25,000 cycles\cite{the_joule_paper}.
@@ -64,7 +73,7 @@
When operated in the nanobeam diffraction mode, where the Bragg diffraction disks do not overlap in the resultant CBED pattern, the relative position of the diffraction disks can be used to quantify the lattice parameter of the section of the TEM sample being illuminated by the electron beam\cite{colin_review}. Since this quantification is being performed at every single individual scan position, this quantification is thus free from drift distortion effects. When extended across the entire field of view, this quantification thus tracks the lattice parameter variations, aka strain, with the precision that cannot be beaten by conventional STEM imaging. Indeed, 4D-STEM has been used for picometer precision strain quantification in two-dimensional crystals, semiconductor heterojunctions, and catalyst nanoparticles \cite{nbed_strain1,nbed_strain2, 4dstem_nanoparticles,yimo_strain,holo_strain}. Most notably, it has been demonstrated that the errors in 4D-STEM measurement are significantly lower than the errors from even drift-corrected annular dark field (ADF) STEM imaging, even when looking at the same particle\cite{4dstem_nanoparticles}.
However, the second issue for electron microscopy-based lattice quantification is present in both conventional STEM and 4D-STEM imaging. This is because lattice parameter quantifications can be performed when the region is imaged is oriented along a low-index crystallographic zone axis, with the problem being absent only for two-dimensional crystals \cite{4dstem_nanoparticles, yimo_strain, holo_strain,disk_registration, nbed_strain2}. Thus, almost every electron microscopy-based lattice parameter variation studies report results from a single particle. Important questions about the statistics of surface strain, the effect of particle size on surface strain have thus remained unanswered. Two recent works have proposed methods to break this logjam -- strain measurements from multiple zone axes, and using cepstrum functions to process CBED datasets, and both of them have shown promising results\cite{elliot_strain,strain_tensor}. However, both these works were performed on bulk TEM samples, and as per the authors' knowledge no study has used 4D-STEM to look at catalyst nanoparticle clusters.
However, the second issue for electron microscopy-based lattice quantification is present in both conventional STEM and 4D-STEM imaging. This is because lattice parameter quantification can be performed when the region is imaged is oriented along a low-index crystallographic zone axis, with the problem being absent only for two-dimensional crystals \cite{4dstem_nanoparticles, yimo_strain, holo_strain,disk_registration, nbed_strain2}. Thus, almost every electron microscopy-based lattice parameter variation studies report results from a single particle. Important questions about the statistics of surface strain, the effect of particle size on surface strain have thus remained unanswered. Two recent works have proposed methods to break this logjam -- strain measurements from multiple zone axes, and using cepstrum functions to process CBED datasets, and both of them have shown promising results\cite{elliot_strain,strain_tensor}. However, both these works were performed on bulk TEM samples, and as per the authors' knowledge no study has used 4D-STEM to look at catalyst nanoparticle clusters.
In this work, we present an approach to compare the unit cell size of nanoparticles even when they are not oriented along a low-index crystallographic axis. Using this approach, we demonstrate that multiple nanoparticles' unit cell size variations can be visualized from a single 4D-STEM dataset. We then use this technique to look at both beginning-of-life (BOL) and end-of-life (EOL) PtCo nanoparticles.
@@ -78,7 +87,7 @@
\end{figure*}
\section*{Author Contributions}
D.M., D.A.C. and M.J.Z. designed the study. C.W. and J.S. grew the nanoparticles. M.J.Z. performed the 4D-STEM experiments. D.M. developed the 4D-STEM strain directed the data analysis, developed the quantification algorithms, analyzed the 4D-STEM data and wrote the manuscript primarily, with contributions from other authors. H.Y. developed the particle picking algorithms with D.M. J.A.H. developed the iterative post-specimen aberration corrector and wrote the Python codes for the same.
D.M., D.A.C. and M.J.Z. designed the study. C.W. and J.S. grew the nanoparticles. M.J.Z. performed the 4D-STEM experiments. D.M. developed the 4D-STEM strain directed the data analysis, developed the quantification algorithms, analyzed the 4D-STEM data and wrote the manuscript, with contributions from other authors. H.Y. and D.A.C. developed the particle picking algorithms. J.D.H. developed the iterative post-specimen aberration corrector and wrote the Python codes for the same.
\caption{\label{fig:all_points}\textbf{Combining all diffraction patterns. a,} Radial location of all the diffraction points with respect to the central diffraction disk. There are a total of two datasets - beginning-of-life (BOL) and end-of-life (EOL). \textbf{b,} Expressing the diffraction points in \autoref{fig:all_points}\blu{a} in polar co-ordinates. \textbf{c,} Probability of locating a diffraction point at a certain radial distance from the center, obtained from \autoref{fig:all_points}\blu{b}.}
\caption{\label{fig:linear_corr}\textbf{Linear Correction. a,} Subset of points linearly fitted, shown in cartesian co-ordinates. \textbf{b,} Points shown in \autoref{fig:linear_corr}\blu{a} shown in polar co-ordinates, where the waviness arising due to aberrations are clearly visible.}
\end{figure}
However, there is a problem - cleanly visible in \autoref{fig:all_points}\blu{b}. There are fundamentally five peaks - and the radial distances of those peaks should be pretty constant with respect to the angle. However, there is a waviness in the data, which broadens the peaks and makes the data very difficult to interpret. Like, look at the peak at $\mathrm{\approx70}$ pixels in \autoref{fig:all_points}\blu{c}. It is split into two - because of the waviness. These distortions have been observed before too, and their influence on the data analysis noted by Mahr \textit{et al.}\cite{4dstem_distortions}.
And the source of this waviness is post-specimen lens aberrations, which we have to correct. One way to express the aberrations is through \autoref{eq:ps_aberrations}.
where $r_i^0$ is the real radial distance, $r_i^a$ is the measured radial distance due to aberrations, and $n$ is the aberration order. Thus, one way to think of this is that there are multiple sine waves. The frequency of the sine wave is the aberration order. Also, all the sine waves are not co-located, which is an issue. So, we need to build an aberration corrector function that does this. Secondly, why is this multiplicative? If you compare the peaks at 70 pixels versus those at 100 pixels in \autoref{fig:all_points}\blu{b}, the higher radius peaks are wavier. Thus, the effect is radially dependent, and not a constant, which means multiplicative.
To test the basic nature of aberrations, we chose to do a linear correction based on the modified \autoref{eq:lin_corr}.
The results from following such a correction is demonstrated in \autoref{fig:linear_corr}\blu{a} and \autoref{fig:linear_corr}\blu{b}, where the linear correction was performed only on a subset of the data. This subset belonged to BOL particles, and positions away from particle shell where maximum strain variation is expected. Also, only those points that were between 80 to 100 pixels from the center were chosen, thus ensuring only one peak value was represented.
While this demonstrated that we are correct on the technical nature of the aberrations, this does not correct the whole dataset. So, what we need is to do this with non-linear (multiplicative) correction on the entire dataset. To do this, we need to find the parameters $a_i$ and $r_i$ that best fit the data. This is done using the Expectation-Maximization (EM) algorithm\cite{em_algo}. The EM algorithm is a two-step process. In the first step, we calculate the expectation of the data, given the current parameters. In the second step, we calculate the maximum likelihood of the parameters, given the current expectation. This is repeated until the parameters converge. The expectation is calculated using the following equation.
