  Scanning transmission electron microscopy (STEM) has been extensively used
for imaging complex materials down to atomic resolution. The most commonly
employed STEM modality, annular dark-field imaging, produces
easily-interpretable contrast, but is dose-inefficient and produces little to
no discernible contrast for light elements and weakly-scattering samples. An
alternative is to use STEM phase retrieval imaging, enabled by high speed
detectors able to record full images of a diffracted STEM probe over a grid of
scan positions. Phase retrieval imaging in STEM is highly dose-efficient,
enabling the measurement of the structure of beam-sensitive materials such as
biological samples. Here, we comprehensively describe the theoretical
background, algorithmic implementation details, and perform both simulated and
experimental tests for three iterative phase retrieval STEM methods:
focused-probe differential phase contrast, defocused-probe parallax imaging,
and a generalized ptychographic gradient descent method implemented in two and
three dimensions. We discuss the strengths and weaknesses of each of these
approaches by comparing the transfer of information using analytical
expressions and numerical results for a white-noise model. This presentation of
STEM phase retrieval methods aims to make these methods more approachable,
reproducible, and more readily adoptable for many classes of samples.
