Elastic delocalization in
EELS
Authors: Löffler Stefan 
Published: December 2016

Inelastic delocalization – caused by the long‐ranged Coulomb interaction – is a well‐known phenomenon in EELS that limits the achievable spatial resolution [1]. For low‐loss EELS, it can lead to a spatial resolution of worse than several nanometers. For core‐loss EELS with energy transfers > ~100 eV, on the other hand, it is generally of the order of 1 Ångström and, therefore, generally does not prevent the acquisition of atomically resolved elemental maps.

Another aspect that is often overlooked, however, is the
elastic
delocalization caused by the extent and the elastic scattering of the electron beam itself inside the crystal [2]. With the ever‐improving aberration correctors and, consequently, ever‐increasing convergence angles, this becomes more and more of an issue. Especially when dealing with samples that are not ideal single crystals, e.g., due to inhomogeneities, embedded nanoparticles, or interfaces, the elastic delocalization can become a severe challenge for atomic‐resolution EELS.

In fig. 1, the case of a NdGaO
3
/LaMnO
3
interface is shown for different convergence angles. The propagation was calculated using the multislice approach [3] for an incident beam energy of 300 keV and no spherical aberration. It is clear that even when the beam is nominally positioned well inside one material, parts of it still extend across the interface into the other material [4]. Therefore, in this situation, one will pick up EELS intensity coming from both sides of the interface (assuming a sufficiently large collection angle; for small collection angles, the situation will be complicated further by the elastic scattering of the beam after the inelastic excitation, which may lead to scattering outside the aperture). In addition, elastic scattering complicates the z sensitivity.

There are several ways to circumvent the problem of elastic delocalization. On the one hand, it is possible to use very thin samples for which elastic scattering and beam broadening are less severe. For large convergence angles, however, this limits the thickness to below ~10 nm which, in turn, decreases the total EELS signal due to the reduced number of atoms. On the other hand, as is evident from fig. 1, the influence of the elastic delocalization can also be decreased by
decreas
ing
the convergence angle. While this may seem counter‐intuitive at first, it can significantly decrease the beam broadening, thus reducing spurious signals coming from adjacent columns. In addition, electron‐vortex beams [5] are also a promising candidate for reducing elastic delocalization due to topological protection [6,7,8]. In addition, the vorticity causes their intensity to vanish in the center, giving them their typical donut shape. While this can make ADF images more difficult to interpret as the elastic scattering likelihood has its maximum when the beam is not actually centered on an atomic column, it does not pose a problem for core‐loss EELS for which the probe beam scatters off the sample electrons: as the electron cloud
surrounds
the nuclei, the inelastic scattering likelihood has its maximum when the donut‐shaped beam is on the atomic column.

Elastic delocalization is unavoidable. This work shows possible ways to mitigate its detrimental effects on core‐loss EELS and thereby paves the way for a better interpretation and quantification of atomic‐resolution mapping, especially in the practically relevant cases of non‐homogeneous samples and interfaces.