Dopant site occupancy determined by core-loss-filtered, position-averaged convergent beam electron diffraction

TL;DR

This work shows that core-loss filtered PACBED patterns, when averaged over probe positions, are effectively equivalent to rocking patterns used in ALCHEMI, enabling dopant site occupancy quantification via EELS-like analysis. By incorporating a transition-potential framework and a fractional occupancy model, the authors develop tilt-dependent $k$-factors and demonstrate the necessity of simulations to correct for delocalization differences across elements. Through a spinel MgAl$_2$O$_4$:Fe case study, they show the approach can quantify dopant distribution between Mg and Al sites, but emphasize dose requirements and the limitations of the fractional occupancy approximation, especially at low concentrations or with delocalized transitions. The findings provide a pathway for using energy-filtered 4D-STEM data to perform ALCHEMI-like dopant analyses, with practical guidance on experimental conditions and required doses.

Abstract

In the elastic scattering regime, probe position-averaged convergent beam electron diffraction (PACBED) patterns have proven robust for estimating specimen thickness and mistilt. Through simulation, we show that core-loss-filtered PACBED patterns can be used to measure the site occupancy of a small concentration of dopants in an otherwise known crystal structure. By leveraging the reciprocity between scanning and conventional transmission electron microscopy, we interpret core-loss PACBED patterns using a strategy traditionally used for determining dopant concentrations via energy dispersive X-ray spectroscopy. We show that differences in the interaction range of different elements hinder a purely measurement-based quantification strategy, but that this can be overcome through comparison with simulations that generalize the Cliff-Lorimer k-factors.

Figures (14)

• Figure 1: (a) Core-loss-filtered PACBED experiment wherein a 4D-STEM dataset is averaged over scan positions, with the detector pixel at $-{\bf q}_\perp$ indicated, together with corresponding simulated images for (b) $\alpha=65$mrad and (c) $\alpha=30$mrad probe forming apertures for an iron-doped spinel with Mg-K, Al-K, Fe-L$_1$ ionization and assuming a narrow energy window 10eV above the corresponding edge. (d) Rocking curve CTEM experiment for an incident plane-wave where signal on an EELS detector with semiangle $\alpha$ is recorded as a function of plane-wave illumination wavevector ${\bf q}=({\bf q}_\perp,q_z)$ direction, together with corresponding simulated incoherent channelling patterns (ICPs) for (e) $\alpha=65$mrad and (f) $\alpha=30$mrad, again for an iron-doped spinel with Mg-K, Al-K, Fe-L$_1$ ionization and assuming a narrow energy window 10eV above the corresponding edge. The approximate reciprocity between (a) and (d) explains why the patterns in (b) and (e) are indistinguishable, and likewise the patterns in (c) and (f). (g) Rocking curve CTEM experiment where X-ray (denoted with $\gamma$) signal on an EDX detector is recorded as a function of plane-wave illumination incidence angle. (h) Corresponding simulated ICPs, with very similar appearance to the patterns in (b) and (e).
• Figure 2: Process for simulating core-loss PACBED patterns as performed using $S$-matrices. The incident beam, $\psi_0$ (grey), elastically scatters through the specimen, as described by the operator $S_1$. When the wavefield encounters an atom of suitable core-shell energy, an inelastic wavefield (green or purple, indicating two of the many different transitions contributing) is generated as per \ref{['eq:elastic_to_inelastic']}, which is subsequently propagated elastically through the specimen using $S_2$. The various inelastic diffraction pattern intensities are added as per \ref{['eq:total_intensity']}. (Since the inelastic mean free path for ionization is larger than the sample thicknesses we neglect double inelastic scattering, which would anyway produce a different energy loss.)
• Figure 3: (a) Schematic of a doped generic crystal structure using fractional occupancy within the elastic scattering regime, where each site is partially occupied in proportion to the probability of finding a particular species at that site. (b) Schematic extending the fractional occupancy to the inelastic channels formulation, where the transition potentials, depicted by the spherical harmonic function inset in each of the atoms, are also partitioned based on the probability of finding a particular species at that site.
• Figure 4: Demonstration of spinel magnesium K ICP contrast with crystal thickness. The crystal is required to be thick enough to ensure channelling which produces stronger contrast. ICPs shown correspond to core-loss PACBED with 30mrad probe forming aperture, 33mrad detector semi-opening angle and 300keV accelerating voltage. Sufficient contrast is achieved at around 32nm.
• Figure 5: ICPs of an iron-doped spinel with 5% Fe doping on magnesium sites and 7% on aluminium sites, where the transition potentials were replaced with Gaussians characterized by their standard deviations, where $\sigma_{\rm Fe}=0.5$Å and $\Delta \sigma=\sigma_{\rm Mg/Al} - \sigma_{\rm Fe} \geq 0$Å. A 20mrad probe forming aperture and a 30mrad detector are assumed. (a) $\Delta\sigma=0.0$Å, corresponding to an exact linear model with unique solution corresponding to the correct doping concentration. (b) $\Delta\sigma=0.2$Å, corresponding to a failing linear model with unique least-squares solution. (c) $\Delta\sigma=0.8$Å, corresponding to a failed linear model with many least-squares solutions. Observe that the core-loss PACBEDs increasingly resemble the (d) elastic PACBED with increasing delocalization.