Coplanarity Index for Potential Slip Systems Predicting in Plastic Semiconductors

Abstract

Despite growing interest in room-temperature plastic semiconductors, understanding and predicting slip modes in semiconductors remains a fundamental challenge due to their complex, low-symmetry structures. Applying analytical frameworks of metal plasticity to semiconductors directly can lead to the ignorance of the quasi-coplanarity in semiconductors due to their complex structures. In this work, we introduce a geometric-descriptor guided strategy to identify potential operative slip systems in semiconductors. A coplanarity index (CI) is developed to quantitatively characterize the quasi-aligned nature of atomic planes. The applicability of the CI is demonstrated in high-performance plastic thermoelectric material Mg₃Sb₂. A quasi-aligned pyramidal plane \left\{0\overline{1}11\right\} with high CI value in hexagonal Mg₃Sb₂ is predicted to be the active slip plane for 〈c + a〉 dislocation slip, accommodating plastic deformation along the c-axis, which is a mechanism that has not been previously recognized in hexagonal semiconductors. The CI values of the \left\{0\overline{1}11\right\} planes also reflect the differences in the atom configurations among the isostructure AB₂X₂ Zintl phases, which indicates the good plasticity of Mg₃Sb₂ from the perspective of coplanarity. This geometry-energy integrated framework offers a quantitative methodology for understanding and designing ductile functional semiconductors.

Keywords

coplanarity index / geometric descriptor / plastic semiconductor / slip system prediction

References

[1] T. Deng, Z. Gao, Z. Li, et al., “Room-Temperature Exceptional Plasticity in Defective Bi₂Te₃-Based Bulk Thermoelectric Crystals,” Science 386, no. 6726 (2024): 1112–1117.

[2] T. R. Wei, M. Jin, Y. Wang, et al., “Exceptional Plasticity in the Bulk Single-Crystalline van der Waals Semiconductor InSe,” Science 369, no. 6503 (2020): 542–545.

[3] X. Shi, H. Chen, F. Hao, et al., “Room-Temperature Ductile Inorganic Semiconductor,” Nature Materials 17, no. 5 (2018): 421–426.

[4] L. Qiu, S. Wang, X. Zhou, et al., “Phase Transformation Induced Plastic Deformation Mechanism in α2-Ti₃Al,” Interdisciplinary Materials 4, no. 3 (2025): 524–534.

[5] J. Luo, J. Chen, Z. Gao, et al., “Ductile Inorganic Ferromagnetic Semiconductor,” Advanced Materials 38, no. 4 (2026): e14083.

[6] X. Li, Y. Meng, W. Li, et al., “Multislip-Enabled Morphing of All-Inorganic Perovskites,” Nature Materials 22, no. 10 (2023): 1175–1181.

[7] Y. Wang, A. Li, Y. Hong, et al., “Iterative Sublattice Amorphization Facilitates Exceptional Processability in Inorganic Semiconductors,” Nature Materials 24, no. 10 (2025): 1545–1553.

[8] H. Hu, Y. Wang, C. Fu, X. Zhao, and T. Zhu, “Achieving Metal-Like Malleability and Ductility in Ag₂Te_1−xS_x Inorganic Thermoelectric Semiconductors With High Mobility,” Innovation (Cambridge (Mass.)) 3, no. 6 (2022): 100341.

[9] Z. Gao, S. Yang, Y. Ma, et al., “Warm Metalworking for Plastic Manufacturing in Brittle Semiconductors,” Nature Materials 24 (2025): 1538–1544.

[10] Z. Yao, Z. Guan, S. Zhang, et al., “Towards Energy-Autonomous Wearables: Multimodal Harvesting Technologies and Sustainable Applications,” Innovation Materials 3, no. 3 (2025): 100143.

[11] Y. Liu, Q. Zhang, A. Huang, et al., “Fully Inkjet-Printed Ag₂Se Flexible Thermoelectric Devices for Sustainable Power Generation,” Nature Communications 15, no. 1 (2024): 2141.

[12] Z. Li, T. Deng, P. Qiu, et al., “Plasticity of Bi₂Te₃-Family Thermoelectric Crystals,” Nature Communications 16, no. 1 (2025): 5190.

[13] J. Lubliner, Plasticity Theory (Pearson Education, 2008).

[14] W. Zhang, R. Yu, K. Du, Z. Cheng, J. Zhu, and H. Ye, “Undulating Slip in Laves Phase and Implications for Deformation in Brittle Materials,” Physical Review Letters 106, no. 16 (2011): 165505.

[15] X. Dong, L. Jiao, H. Tong, et al., “Heavy Doping Driven Ultra-Low Lattice Thermal Conductivity and Mechanical Enhancement in Mg₃(Bi, Sb)₂ for Efficient Power Generation,” Acta Materialia 300 (2025): 121451.

[16] Z. Liu, N. Sato, W. Gao, et al., “Demonstration of Ultrahigh Thermoelectric Efficiency of ∼7.3% in Mg₃Sb₂/MgAgSb Module for Low-Temperature Energy Harvesting,” Joule 5, no. 5 (2021): 1196–1208.

[17] L. Wang, A. Li, X. Wu, J. Li, and T. Mori, “Leveraging Carrier Mobility Enables High-Performance Mg₃(Sb, Bi)₂ Thermoelectrics,” National Science Review 13, no. 1 (2026): nwaf507.

[18] J. Li, S. Zhang, F. Jia, et al., “Point Defect Engineering and Machinability in n-Type Mg₃Sb₂-Based Materials,” Materials Today Physics 15 (2020): 100269.

[19] P. Zhao, W. Xue, Y. Zhang, et al., “Plasticity in Single-Crystalline Mg₃Bi₂ Thermoelectric Material,” Nature 631 (2024): 777–782.

[20] A. Li, Y. Wang, Y. Li, et al., “High Performance Magnesium-Based Plastic Semiconductors for Flexible Thermoelectrics,” Nature Communications 15, no. 1 (2024): 5108.

[21] M. Hu, J. Yang, Y. Wang, et al., “Helical Dislocation-Driven Plasticity and Flexible High-Performance Thermoelectric Generator in α-Mg₃Bi₂ Single Crystals,” Nature Communications 16, no. 1 (2025): 128.

[22] T. Zhang, J. Yan, J. Ke, et al., “Substantial Planar Plastic Anisotropy in Inorganic Mg₃Bi₂ Single Crystals,” Proceedings of the National Academy of Sciences 122, no. 47 (2025): e2516449122.

[23] D. L. Medlin, N. Yang, C. D. Spataru, L. M. Hale, and Y. Mishin, “Unraveling the Dislocation Core Structure at a van der Waals Gap in Bismuth Telluride,” Nature Communications 10, no. 1 (2019): 1820.

[24] P. Chowdhury and H. Sehitoglu, “A Revisit to Atomistic Rationale for Slip in Shape Memory Alloys,” Progress in Materials Science 85 (2017): 1–42.

[25] Z. Wu, R. Ahmad, B. Yin, S. Sandlöbes, and W. A. Curtin, “Mechanistic Origin and Prediction of Enhanced Ductility in Magnesium Alloys,” Science 359, no. 6374 (2018): 447–452.

[26] Z. Wu and W. A. Curtin, “Mechanism and Energetics of <c + a> Dislocation Cross-Slip in hcp Metals,” Proceedings of the National Academy of Sciences of the United States of America 113, no. 40 (2016): 11137–11142.

[27] C. A. Volkert and E. T. Lilleodden, “Size Effects in the Deformation of Sub-Micron Au Columns,” Philosophical Magazine 86, no. 33–35 (2006): 5567–5579.

[28] Z. Zhang, J. Zhang, J. Wang, et al., “Toward the Development of Mg Alloys With Simultaneously Improved Strength and Ductility by Refining Grain Size Via the Deformation Process,” International Journal of Minerals, Metallurgy, and Materials 28, no. 1 (2020): 30–45.

[29] B. Joós and M. S. Duesbery, “The Peierls Stress of Dislocations: An Analytic Formula,” Physical Review Letters 78 (1997): 266–269.

[30] T. W. Clyne and J. E. Campbell Testing of the Plastic Deformation of Metals. T. W. Clyne ed. 2021.

[31] X. L. Nan, H. Y. Wang, L. Zhang, J. B. Li, and Q. C. Jiang, “Calculation of Schmid Factors in Magnesium: Analysis of Deformation Behaviors,” Scripta Materialia 67, no. 5 (2012): 443–446.

[32] Y. Bu, Z. Li, J. Liu, H. Wang, D. Raabe, and W. Yang, “Nonbasal Slip Systems Enable a Strong and Ductile Hexagonal-Close-Packed High-Entropy Phase,” Physical Review Letters 122, no. 7 (2019): 075502.

[33] B. Silvi and A. Savin, “Classification of Chemical Bonds Based on Topological Analysis of Electron Localization Functions,” Nature 371 (1994): 683–686.

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2026 The Author(s). Interdisciplinary Materials published by Wuhan University of Technology and John Wiley & Sons Australia, Ltd.