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An atomic Boltzmann machine capable of self-adaption


The quest to implement machine learning algorithms in hardware has focused on combining various materials, each mimicking a computational primitive, to create device functionality. Ultimately, these piecewise approaches limit functionality and efficiency, while complicating scaling and on-chip learning, necessitating new approaches linking physical phenomena to machine learning models. Here, we create an atomic spin system that emulates a Boltzmann machine directly in the orbital dynamics of one well-defined material system. Utilizing the concept of orbital memory based on individual cobalt atoms on black phosphorus, we fabricate the prerequisite tuneable multi-well energy landscape by gating patterned atomic ensembles using scanning tunnelling microscopy. Exploiting the anisotropic behaviour of black phosphorus, we realize plasticity with multi-valued and interlinking synapses that lead to tuneable probability distributions. Furthermore, we observe an autonomous reorganization of the synaptic weights in response to external electrical stimuli, which evolves at a different time scale compared to neural dynamics. This self-adaptive architecture paves the way for autonomous learning directly in atomic-scale machine learning hardware.

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Fig. 1: Neural dynamics from coupled cobalt atoms on BP.
Fig. 2: Construction of a binary atomic synapse via anisotropic coupling.
Fig. 3: Multi-valued synapses.
Fig. 4: Synaptic dynamics and self-adaption.

Data availability

The data from this work can be obtained from the corresponding author upon reasonable request.


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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 818399). This research was funded in part by ONR grant no. N00014-17-1-2569. A.A.K. and E.J.K. acknowledge the NWO-VIDI project ‘Manipulating the interplay between superconductivity and chiral magnetism at the single-atom level’ with project no. 680-47-534. B.K. acknowledges NWO-VENI project ‘Controlling magnetism of single atoms on black phosphorus’ with project no. 016.Veni.192.168.

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Authors and Affiliations



B.K. and E.J.K. performed the experiments under the direction and supervision of A.A.K. B.K. and E.J.K. developed the data analysis, while B.K., E.J.K., H.J.K. and A.A.K. participated in the scientific analysis. W.M.J.v.W. developed the a.c. experimental setup. H.J.K. performed the Boltzmann machine modelling. A.A.K. and H.J.K. designed the experiments. The manuscript was written by B.K., E.J.K., H.J.K. and A.A.K.

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Correspondence to Alexander A. Khajetoorians.

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The authors declare no competing interests.

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Peer review information Nature Nanotechnology thanks Giuseppe Carleo, Matthew Ellis and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Figs. 1–12, Discussion and Tables 1–3.

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Kiraly, B., Knol, E.J., van Weerdenburg, W.M.J. et al. An atomic Boltzmann machine capable of self-adaption. Nat. Nanotechnol. 16, 414–420 (2021).

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