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. 2021 Jun 10;12:3524. doi: 10.1038/s41467-021-23844-z

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 4 — Surface states and the evolution of the hinge states through the top surface a–e and the side surface f–j for FM axion insulator MnBi₂Te₄. a Top surface states. b–d LDOS of a sample with 20-unit-cell thick in the x-direction, including the right hinge b, center of the top surface c, and the left hinge d. e Intensity of the in-band (blue) and in-gap (red) hinge states projected onto different positions of the top surface as a function of the unit-cell index (#1 and #20 are the hinges, while #10 is the center of the computed cell). f Side surface states. g–i LDOS of a sample with 30-unit-cell thick in the z-direction (i.e., 30 VdW layers), including the top hinge g, center of the side surface h, and the bottom hinge i. j Similar to e but for the side surface.