The sub-angstrom electron probe of an aberration-corrected scanning transmission electron microscope (STEM) can move individual dopant atoms and assemble few-atom structures in crystals, most prominently silicon dopants in graphene. However, the demonstrated manipulations are stochastic and only partially understood: beam-induced transformations proceed through many weakly characterized elementary steps, defect modifications were described as 'as yet, uncontrollabl[e]' in the founding demonstrations, and controllers (human or machine-learned) navigate statistical transition rates rather than deterministic dynamics. The problem, formulated by this literature, is to achieve reliable, deterministic atom-by-atom fabrication with the STEM beam, which requires a quantitative understanding of electron-beam-induced transformation mechanisms at high spatiotemporal resolution together with control strategies robust to the intrinsic stochasticity. Later work extended directed dopant motion in graphene, quantified manipulation mechanisms in specific systems, automated single-defect transformations, and demonstrated machine-learned navigation of a single silicon atom to target sites; in 2026, deterministic atomic engineering was achieved in one favorable system (steering native Cr interstitials in CrSBr to user-defined positions, creating more than 40,000 designed defects), establishing that determinism is attainable in principle. The surviving open core is deterministic, repeatable fabrication of user-specified atomic structures in covalent two-dimensional lattices — substitutional dopant placement and heterogeneous multi-atom assembly — supported by a predictive, transferable theory of electron-beam-induced single-atom transition rates.

Background

A scanning transmission electron microscope (STEM) raster-scans a beam of electrons focused to a probe smaller than the spacing between atoms. Such sub-angstrom probes became possible only after aberration correction — the compensation of the lens imperfections that would otherwise blur the electron beam — and they let the microscope both image individual atoms and act on them: when the beam transfers enough energy to an atom (above the so-called knock-on threshold, set by the accelerating voltage and the collision geometry), the atom is displaced from its site; near or below threshold, the beam can still activate hops of impurity atoms between lattice sites, break or form bonds, and drive local atomic-scale chemistry. This turns the microscope into an actuator with atomic resolution, in contrast to scanning tunneling microscopy (STM), the earlier workhorse of atom manipulation, which uses forces from a sharp metal tip but generally requires cryogenic temperatures, conducting surfaces, and manipulates about one atom at a time.

The paradigm system is substitutional dopants — foreign atoms replacing host atoms in a lattice — in two-dimensional (2D) materials, especially silicon dopants in graphene (a one-atom-thick sheet of carbon in a hexagonal lattice). In 2017–2018 it was demonstrated that the STEM beam can place single Si atoms into graphene vacancies, passivate vacancies by Si substitution, move Si dopants from site to site, and assemble Si dimers, trimers, and larger clusters, with the individual atomic steps captured in time-series imaging; surveys of this capability and its limits had already been compiled by the electron-irradiation community (Susi, Meyer and Kotakoski, Ultramicroscopy 180, 163, 2017). Unlike STM, this manipulation works at room temperature in semiconducting and insulating 2D lattices and images the system while acting on it.

The obstacle is that every manipulation is a probabilistic scattering event. The probability that a given atom moves depends on sub-angstrom details of where the probe sits relative to the lattice, on the accelerating voltage relative to displacement thresholds, on accumulated electron dose (electrons delivered per unit area), and on sample temperature; competing channels such as etching, unwanted defect creation, and atom ejection operate simultaneously. The founding papers stated the limitation explicitly: defect modifications occurred 'as yet, uncontrollably' (Dyck, Kim, Kalinin and Jesse, Appl. Phys. Lett. 111, 113104, 2017); the process 'proceeds via a large number of weakly-understood individual steps' with 'high stochasticity of the environment' (Vasudevan, Ghosh, Ziatdinov and Kalinin, Nanotechnology 33, 115301, 2022); and the inability to resolve atomic transformation pathways at high spatiotemporal resolution — atomic spatial resolution combined with millisecond-or-faster temporal resolution — 'has limited widespread use' of directed atomic fabrication (Boebinger et al., npj Computational Materials 10, 260, 2024). Since then, machine-learning tools have been brought to bear: deep-network decoding of fast, noisy image streams at up to about 100 frames per second, automated beam-control workflows acting on specific atomic configurations, reinforcement-learning controllers trained in simulated environments, and learned transition-dynamics models used to guide a single Si atom across the graphene lattice to chosen targets (Schwarzer et al., Adv. Mater. Interfaces 12, 2025). Nevertheless, reviews as late as 2024 describe atomic fabrication as 'still in its infancy' with dynamics that 'remain elusive' (Zhou et al., Nanomaterials 14, 1718, 2024). In 2026, Klein and coworkers achieved deterministic atomic engineering in one favorable system — steering native chromium interstitials (Cr atoms sitting between regular lattice sites) in the layered magnetic semiconductor CrSBr into user-selected positions, creating more than 40,000 designed vacancy–interstitial defects — proving that deterministic control is attainable in principle, but by a mechanism specific to that material and defect type, not for substitutional dopants or heterogeneous structure assembly in covalent 2D lattices.

Problem Statement

Achieve reliable, deterministic atom-by-atom fabrication using the sub-atomically focused electron beam of an aberration-corrected scanning transmission electron microscope. Specifically: given a specified atomic-scale target — individual dopant atoms placed at selected lattice sites, and/or multi-atom structures (dimers, trimers, extended clusters, defect lines) of defined geometry and composition — in a host crystal (paradigmatically a covalent two-dimensional lattice such as graphene), establish (i) a quantitative, predictive understanding of the electron-beam-induced transformation mechanisms involved, that is, the single-atom transition probabilities as functions of beam position relative to the lattice, accelerating voltage, dose and dwell schedule, and sample temperature, resolved at the high spatiotemporal resolution needed to identify the elementary steps; and (ii) beam-control and feedback strategies (manual, automated, or machine-learned) that use this understanding to drive the system from an initial configuration to the target structure reproducibly and with quantified success rates approaching deterministic control, with every fabrication step verified by in-situ atomic-resolution imaging. The problem is solved for a material system when repeated fabrication of user-specified structures succeeds with predictable near-unit probability — a per-fabrication-step and per-target-structure success probability approaching one, stated explicitly and demonstrated over a stated number of repeated trials, where ‘deterministic control’ denotes such reproducible near-unit success rather than the elimination of the intrinsic stochasticity of beam-induced transitions — and the governing transition rates are quantitatively explained to a stated, validated accuracy; it is solved in general when this holds across dopant species, host lattices, and structure classes — in particular beyond favorable native-defect systems such as Cr interstitial steering in CrSBr, to substitutional dopants and heterogeneous multi-atom assembly in two-dimensional materials — rather than for a single defect type in a single host. Resolution of the open problem identified above corresponds to this general tier restricted to its surviving open core — deterministic, repeatable fabrication of user-specified atomic structures in covalent two-dimensional lattices — and therefore requires each of the two structure classes — substitutional dopant placement and heterogeneous multi-atom assembly — to be achieved across at least two host lattices and at least two dopant species within covalent two-dimensional lattices, supported by a predictive, transferable theory of electron-beam-induced single-atom transition rates whose parameters, fixed on one system (a single host lattice and dopant species), predict the single-atom transition rates of a dopant species or host lattice not used in fitting, within a stated accuracy; it does not require every host lattice or structure class, and achieving the per-system tier in an additional system constitutes major partial progress toward resolution rather than resolution itself.

References

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