Non-monotonic magnetic friction from collective rotor dynamics

Nature Materials (2026)Cite this article Amontons’ law postulates a monotonic relationship between frictional force and the normal load applied to a sliding contact. This empirical rule, however, fails in systems where internal degrees of freedom—such as structural or electronic order—play a central role. Here, we demonstrate that friction can emerge entirely from magnetically driven configurational dynamics. Using a two-dimensional array of rotatable magnetic moments sliding over a commensurate magnetic substrate, we observe a pronounced non-monotonic dependence of friction on the interlayer separation, and thus on the effective load. The friction peaks at an intermediate distance where competing ferromagnetic and antiferromagnetic interactions induce dynamical frustration and hysteretic torque cycles during sliding. Molecular dynamics simulations and a simplified two-sublattice model confirm that energy dissipation is governed by collective magnetic reorientations and their hysteresis. Our results establish scale-free sliding-induced changes in interfacial collective magnetic order, which has a strong impact on friction, and thus open new possibilities for contactless friction control, magnetic sensing and the design of reconfigurable, wear-free frictional interfaces and metamaterials.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutThe data supporting the findings of this study are available within the Article and the Supplementary Information, and from the Zenodo data repository60 at https://doi.org/10.5281/zenodo.18487116.The source code of the simulation is available on Zenodo61 via https://doi.org/10.5281/zenodo.18487550.Desplanques, Y. Amontons–Coulomb friction laws, a review of the original manuscript. SAE Int. J. Mater. Manuf. 8, 98–103 (2015).Article Google Scholar Molina-Sánchez, A. et al. Effect of spin-orbit interaction on the optical spectra of single-layer, double-layer, and bulk MoS2. Phys. Rev. B 88, 045412 (2013).Article Google Scholar Sun, J. et al. Superlubricity enabled by pressure-induced friction collapse. J. Phys. Chem. Lett. 9, 2554–2559 (2018).Article CAS PubMed Google Scholar Deng, Z., Smolyanitsky, A., Li, Q., Feng, X.-Q. & Cannara, R. J. Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale. Nat. Mater. 11, 1032–1037 (2012).Article CAS PubMed Google Scholar Hu, Q. et al. Ferrielectricity controlled widely-tunable magnetoelectric coupling in van der Waals multiferroics. Nat. Commun. 15, 3029 (2024).Article CAS PubMed PubMed Central Google Scholar Gao, X., Ouyang, W., Hod, O. & Urbakh, M. Mechanisms of frictional energy dissipation at graphene grain boundaries. Phys. Rev. B. 103, 045418 (2021).Article CAS Google Scholar Gao, X., Ouyang, W., Urbakh, M. & Hod, O. Superlubric polycrystalline graphene interfaces. Nat. Commun. 12, 5694 (2021).Article CAS PubMed PubMed Central Google Scholar Benassi, A., Vanossi, A., Santoro, G. E. & Tosatti, E. Sliding over a phase transition. Phys. Rev. Lett. 106, 256102 (2011).Article CAS PubMed Google Scholar Panizon, E. et al. Friction anomalies at first-order transition spinodals: 1T-TaS2. New J. Phys. 20, 023033 (2018).Article Google Scholar Liu, K. et al. Negative differential friction coefficients of two-dimensional commensurate contacts dominated by electronic phase transition. Nano Res. 15, 5758–5766 (2022).Article CAS Google Scholar Wolter, B. et al. Spin friction observed on the atomic scale. Phys. Rev. Lett. 109, 116102 (2012).Article PubMed Google Scholar Komatsu, H. Model of magnetic friction obeying the Dieterich–Ruina law in the steady state. Phys. Rev. E 100, 052130 (2019).Article CAS PubMed Google Scholar Heinrich, A. Magnetic friction is a sticky business. Phys. Mag. 5, 102 (2012).

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AntonLueders/MagneticFriction: MagneticFriction_v2.0.3. Zenodo https://doi.org/10.5281/zenodo.18487551 (2026).Download referencesWe acknowledge stimulating discussions with A. Vanossi, D. Bossini and V.-L. Heuthe and thank T. Franosch for his input regarding the calculation of the energy transfer rates. We thank D. Barth and J. Steindl for advice and technical assistance in building the force-testing set-up used in this study. We also thank Y. Tang and L. Carls for their assistance in this research. We acknowledge support by the local computing resources through the core facility SCCKN. C.B. acknowledges financial supported by the ERC advanced grant BRONEB (101141477). H.G. thanks Hong Kong Research Grants Council for their support (project 26217625).These authors contributed equally: Hongri Gu, Anton Lüders.Division of Integrative Systems and Design, Hong Kong University of Science and Technology, Hong Kong, ChinaHongri GuDepartment of Physics, University of Konstanz, Konstanz, GermanyHongri Gu, Anton Lüders & Clemens BechingerDepartment of Theoretical Physics, Universität Innsbruck, Innsbruck, AustriaAnton LüdersSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarH.G., A.L. and C.B. came up with the idea and designed the project. H.G. conducted the experiments. A.L. performed the simulations and the analytical modelling. All authors participated in the discussion, analysis and writing of the manuscript.Correspondence to Hongri Gu, Anton Lüders or Clemens Bechinger.The authors declare no competing interests.Nature Materials thanks Alessandro Siria and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary Sections 1–19.Experimental measurements and video tracking, showing the different dynamics of the magnetic moments in the three observed regimes. To visualize the orientation of the moments, the ring-shaped magnets of the slider are encased in a cuboidal plastic casing with differently coloured sides. First the FM regime at h = 6.5 mm is shown. Then the AFM regime is represented by the measurement corresponding to h = 11 mm. Finally, the CP regime at h = 8.5 mm is presented.Direct comparison between the experiments and the simulations. The video compares the results of the simulations with the experimental measurements using h = 6.5 mm, h = 11 mm and h = 8.5 mm as examples for the FM, the AFM and the CP regime, respectively. To visualize the orientation of the moments in the experiments, the ring-shaped magnets of the slider are encased in a cuboidal plastic casing with differently coloured sides. The simulations and the experiments are in excellent agreement.Comparison of the dynamics of the magnetic moments at different heights h. First, the results of the molecular dynamics simulations are shown. Afterwards, the videos of the experimental set-up are presented. To visualize the orientation of the moments in the experiments, the ring-shaped magnets of the slider are encased in a cuboidal plastic casing with differently coloured sides. The moments transition from a state of synchronized motion (FM regime) to the AFM state when h is increased. At the transition, we find the CP regime, where parallel and antiparallel alignments alternate.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsGu, H., Lüders, A. & Bechinger, C. Non-monotonic magnetic friction from collective rotor dynamics. Nat. Mater. (2026). https://doi.org/10.1038/s41563-026-02538-1Download citationReceived: 12 August 2025Accepted: 10 February 2026Published: 18 March 2026Version of record: 18 March 2026DOI: https://doi.org/10.1038/s41563-026-02538-1Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative