Fully nonlocal inelastic scattering computations for spectroscopical transmission electron microscopy methods

The complex interplay of elastic and inelastic scattering amenable to different levels of approximation constitutes the major challenge for the computation and hence interpretation of TEM-based spectroscopical methods. The two major approaches to calculate inelastic scattering cross sections of fast electrons on crystals—Yoshioka-equations-based forward propagation and the reciprocal wave method—are founded in two conceptually differing schemes—a numerical forward integration of each inelastically scattered wave function, yielding the exit density matrix, and a computation of inelastic scattering matrix elements using elastically scattered initial and final states (double channeling). Here, we compare both approaches and show that the latter is computationally competitive to the former by exploiting analytical integration schemes over multiple excited states. Moreover, we show how to include full nonlocality of the inelastic scattering event, neglected in the forward propagation approaches, at no additional computing costs in the reciprocal wave method. Detailed simulations show in some cases significant errors due to the $z$-locality approximation and hence pitfalls in the interpretation of spectroscopical TEM results.

Figure 1

Simulations of Ti $L_3$ edge energy-filtered high-resolution TEM images of ${\mathrm{SrTiO}}_3$ in [110] zone axis orientation for a range of defoci and spherical aberrations. The upper row represents simulations with full nonlocality of the inelastic scattering, while in the bottom we have applied the $z$-locality approximation (similar to Forbes et al. [27]). The position of atomic columns is marked in panels with zero defocus and spherical aberration, at thickness 0.6 nm. Ti columns are marked in green color, Sr columns in orange color, and O columns are marked with smaller spheres of blue color. Dimensions of each individual plot are $5.52\AA \times 3.91\AA$. Blue frames mark thicknesses, where the largest differences between fully nonlocal and $z$-local treatment can be observed; see text for more details.

Figure 2

Histogram of wave vector elongations $\gamma$ for a plane wave of energy 200 keV, incoming along the $\left[110\right]$ direction, scattering on ${\mathrm{SrTiO}}_3$ crystal. Vertical blue solid lines mark the multiples of $c^{★}$ axis, the reciprocal-lattice vector of ${\mathrm{SrTiO}}_3$ [110] supercell, and the light blue shaded areas around them denote the $\pm q_{\mathrm{\Delta }E}$ region. Note that in the first Brillouin zone there are plenty of $\gamma$'s that are in size comparable to $q_{\mathrm{\Delta }E}$.

Figure 3

Periodic channeling of the wave ${\psi }^i$ elastically scattered on ${\mathrm{SrTiO}}_3$ and ${\mathrm{PbZrO}}_3$. In ${\mathrm{SrTiO}}_3$ the wavelength of the channeling amounts to approximately 11 nm at the SrO column, 27 nm at the Ti column. Note the suboscillation with maxima at approximately 10 and 19 nm at the Ti column. In ${\mathrm{PbZrO}}_3$ periodic channeling (with a wavelength of approximately 7 nm) is only observed at the ZrO column, whereas the scattering power of the Pb column quickly disperses the focusing effect beyond the first maxima.

Figure 4

Defocus series of EF-HRTEM images of ${\mathrm{PbZrO}}_3$, calculation for the Pb edges $N_{6,7}$ and $N_{4,5}$, Zr $M_{4,5}$ edge, and O $K$ edge. The upper panels show calculations with full nonlocality of the inelastic interaction, while the lower panels show calculations with $z$-locality approximation. The range of defoci is from $-8$ nm to $+8$ nm, as in Fig. 1. The thicknesses vary from 5 to 35 nm with 5-nm steps. Positions of atomic columns are marked by orange (Pb), green (Zr), and light blue (O) spheres in panels with zero defocus and thickness of 5 nm. Dimensions of each individual plot are $4.18\AA \times 4.18\AA$. Blue frames mark thicknesses, where the largest differences between fully nonlocal and $z$-local treatment can be observed; see text for more details.