Exact inversion of partially coherent dynamical electron scattering for picometric structure retrieval

The greatly nonlinear diffraction of high-energy electron probes focused to subatomic diameters frustrates the direct inversion of ptychographic data sets to decipher the atomic structure. Several iterative algorithms have been proposed to yield atomically-resolved phase distributions within slices of a 3D specimen, corresponding to the scattering centers of the electron wave. By pixelwise phase retrieval, current approaches do not only involve orders of magnitude more free parameters than necessary, but also neglect essential details of scattering physics such as the atomistic nature of the specimen and thermal effects. Here, we introduce a parametrized, fully differentiable scheme employing neural network concepts which allows the inversion of ptychographic data by means of entirely physical quantities. Omnipresent thermal diffuse scattering in thick specimens is treated accurately using frozen phonons, and atom types, positions and partial coherence are accounted for in the inverse model as relativistic scattering theory demands. Our approach exploits 4D experimental data collected in an aberration-corrected momentum-resolved scanning transmission electron microscopy setup. Atom positions in a 20 nm thick PbZr0.2Ti0.8O3 ferroelectric are measured with picometer precision, including the discrimination of different atom types and positions in mixed columns.

following.The first important improvement is using a multislice interaction model.It splits the specimen into multiple slices and calculates their projected potentials V 1 , . . ., V S to obtain M (ϕ, V 1 , . . ., V S ) = e iσV S • (F ⊗ (e iσV S−1 . . .(F ⊗ (e iσV1 • ϕ)))). (S.1) Here, F is the Fresnel propagator, which propagates the wave by the distance between the current and the next slice.The propagator multiplies the wave in the Fourier domain with P (k, ∆z, θ) = e −iπλk 2 ∆z+2πi∆z(kx tan θx+ky tan θy) .(S.2) Here, θ x , θ y is the specimen tilt and ∆z the slice distance.This model describes the coherent forward scattering problem in exact manner, however, established inverse techniques reconstruct the phase of the exponentials without physical constraints so far.
The second improvement is mixed-state ptychography, which addresses partial coherence.As our approach differs significantly from previously used formulations, we discuss it in more detail in the next section.Most important for the work presented here are the methods relying on an atomic model.Furthermore, we chose a different approach to address the limited initial knowledge about the probing wave ϕ.While often the discretized wave is optimized pixelwise, an approach called blind ptychography, we use a semi-blind approach, where we parametrize ϕ by a few aberration parameters.In our results we optimized for defocus C 1 , 2-and 3-astigmatism A 1 and A 2 , coma B 2 and spherical aberration C S .

Supplementary Note 2 Relative tilt between illumination and specimen
All free parameters are optimized via the automatic differentiation routine of Pytorch.For some, like aberration parameters, this is straight forward, as they are used only at a single place, the probe generation.However, other parameters occur in multiple places, most importantly the tilt (another example are atomic positions in models periodized in z direction).For the tilt, we use Θ = (tan θ x , tan θ y ) as free parameteres, and the function that is differentiated can be thought of as Θ → (Θ, . . ., Θ) S−1 times → (P (k, ∆z, Θ), . . ., P (k, ∆z, Θ)).
These S − 1 values are then plugged into the Fresnel propagators F in (S.1).Standard multivariate calculus shows that the gradient induced in Θ is the sum of the gradients induced in the single slices.Note that these gradients scale automatically with the slice thickness, therefore for a given total thickness of the specimen, the learning rate is invariant with respect to different slicing schemes.This approach differes in that regard from the parametrization in [7,8], where only the gradient with respect to the middle slice is considered.

Supplementary Note 3 Partial spatial and temporal coherence
Partial spatial coherence can be modelled by a density function ρ such that Instead of an integral, in simulations one typically chooses a suitable quadrature rule and approximates However, we usually do not know the density ρ exactly and can thus also not pick a suitable quadrature formula for the integral (S.3).That is why we optimize for the weights β k and offsets η k as well during the inversion procedure.The energy spread of the incoming electrons, as well as electron-optical instabilities lead to partial temporal coherence.It can be modelled by letting the defocus of the probe vary slightly in different simulations, and by summing the resulting intensities.The same reasoning as above applies and we end up with an additional offset for the defocus of ϕ.In that respect, these parametrizations allow to estimate the incoherence parameters.
To demonstrate that our approach to partial coherence during the inversion of the problem in physically parametrized manner is reliable, we applied it to the data published by Chen et al. [9].The results are shown below.Negative potential values have been set to zero after a 50 episodes, then the reconstruction continued iterating for 50 more episodes.Also, a regularization of the form ∥FT ∥ 1 , where T is complex-valued phase grating (representing e iσV in an ideal case) was used.This should be contrasted to the phase unwrapping and the so-called missing wedge regularization used by Chen et al. [9] Further adaptations, e.g. also employing the missing wedge regularization, could lead to even better results.
Reconstruction from the dataset of Chen et al. a) Our reconstruction, using 13 offsets for the probes.Reconstruction from Chen et al. is shown for comparison.b) Using just a single probe will not lead to a satisfying reconstruction, demonstrating the necessity of incorporating incoherence.c) shows our probe (calculated from the recovered aberration coefficients, exactly as the algorithm does it) and the first five modes recovered by Chen et al.We use the same probe, up to a defocus and position offset) for each of the 13 incoherently summed exit waves.