Parametrized diffraction inversion by gradient-based optimization employing a fully differentiable multislice

Abstract text (incl. figure legends and references)

With the availability of momentum-resolved STEM data, a multitude of phase retrieval methods were established. Most can be grouped into two categories; direct inversions employing a single object transmission function (OTF) and iterative techniques capable of reconstructing both single- and multiple slice OTFs as well as the STEM probe. Importantly, these optimization schemes act on the pixel-wise distribution of amplitude and phase of OTF and probe.

In fact, the diffraction patterns (DP) correspond to a relatively sparse set [P] of parameters, e.g., aberration coefficients, positions and types of atoms, specimen tilt, coherence parameters and probe positions. The goal of an inversion procedure is the retrieval of [P] iteratively from a starting guess using the gradients with respect to [P]. We introduce a fully differentiable multislice scheme TorchSlice implemented in Pytorch, utilizing GPUs. Gradients are obtained by backpropagation, a method widely used in machine learning. Using backpropagation for inverse multislice was introduced in [1, 2].

Here, we demonstrate the capability of this approach in two steps. First, we employ the availability of gradients to study optimum experimental designs. In particular, a 4D STEM simulation of 2D and bulk MoS2 including vacancies is performed, resembling an experimental data set. A parametrized perfect crystal is then assumed to calculate the gradients with respect to the presence or absence of atoms at their native sites, for different STEM signals, doses, probe foci and aberrations. Fig 1 depicts the success rate for the detection of a vacancy in dependence of dose and focus for the 4D STEM and the STEM bright field signals. It is concluded that 4D STEM is a lot more dose-efficient and robust than BF imaging with respect to vacancy detection.

Secondly, we address the retrieval of bonding effects from 4D STEM data of hBN and GaAs. DFT simulations are used to generate DP including electron redistributions due to chemical bonding. Then, a model based on isolated atoms is set up as initialization for a gradient-based optimization. That allows to evaluate a priori what dose levels and experimental setups are most promising to detect bonding effects. The hBN monolayer required a significantly higher dose level, electrons per DP, while the bonding effect of a 12nm GaAs crystal appeared already at electrons per DP. It is conceivable that an elevated thickness helps to detect bonding effects. Fig 2 depicts the successful reconstruction of bonding effects for GaAs.

Finally, we address further prospects and show first reconstructions of experimental data of 2D heterostructures and ferroelectrics, where we determined the cation displacements from slightly misoriented PZT crystals.

Figure 1: detection probability of a vacancy depending on the dose and the defocus, respectively and from bright field or 4d-stem data, respectively. Probabilities were calculated over 70 runs

Figure 2: Phase grating (PG) in the isolated atom approximation for GaAs on the left; Difference between PG with bonding and PG at the end of reconstruction on the right. The PG was recovered at the area covered by the probes

[1] W. Van den Broek and C. T. Koch, Physical review letters, 2012, 10.1103/PhysRevLett.109.245502

[2] W. Van den Broek and C. T. Koch, Physical Review B, 2013, 10.1103/PhysRevB.87.184108

[3] Support from Helmholtz under contracts VH-NG 1317, ZT-I-0025 and ZT-I-PF-5-028 is gratefully acknowledged.