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. 2021 Dec 10;12:7201. doi: 10.1038/s41467-021-26414-5

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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 — a, b The spectra of the fabricated electric circuit for the hybrid second-order ST effect (a) and SS effect (b). ∈ is eigenvalue. Red, blue and green dots represent the spectra subject to double periodic boundary conditions, x-periodic, y-open boundary conditions, and double open boundary conditions respectively. Comparison with Fig. 2 reveals very good agreement with theory. c, d The voltages (corner, edge, and bulk) with the frequency for the hybrid second-order ST and SS effects. Red, blue and green lines represent the voltages of sites at the corner, edge, and bulk, respectively, with the corner voltage peaking at the resonant frequency as expected from theory (Eq. 2).