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  • Open Access

Symmetry-based design rules for second-harmonic generation in stacked and twisted MoS2 bilayers

Sumanti Patra1,* and Caterina Cocchi1,2,†

  • *Contact author: sumanti.patra@uni-jena.de
  • †Contact author: caterina.cocchi@uni-jena.de
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Phys. Rev. B 114, 225412 – Published 19 October, 2026

DOI: https://doi.org/10.1103/ldws-7yvc

Abstract

Understanding how stacking controls the nonlinear optical response of two-dimensional materials is key to designing van der Waals heterostructures with tailored functionalities. Here, we establish a comprehensive symmetry-based framework mapping the structural configuration of MoS2 bilayers across four point groups (D3h, D3d, C3v, C3) to their second-order susceptibility tensor χ(2). Using group-theory arguments benchmarked against first-principles response-function calculations, we demonstrate how symmetry breaking controls the activation and suppression of individual tensor elements in these systems. We show that the emergence of the in-plane component χxxx in twisted configurations (C3 group) induces a rigid azimuthal rotation of the second-harmonic generation polar lobes, which remains frequency-independent across the entire optical spectrum, locking to half of the structural twist angle. Our findings establish a direct, wavelength-independent optical route for twist-angle determination and provide a clear roadmap for engineering nonlinear optical responses in two-dimensional materials.

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References (44)

  1. Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation, Nano Lett. 13, 3329 (2013).
  2. A. Autere, H. Jussila, Y. Dai, Y. Wang, H. Lipsanen, and Z. Sun, Nonlinear optics with 2D layered materials, Adv. Mater. 30, 1705963 (2018).
  3. S. Klimmer, O. Ghaebi, Z. Gan, A. George, A. Turchanin, G. Cerullo, and G. Soavi, All-optical polarization and amplitude modulation of second-harmonic generation in atomically thin semiconductors, Nat. Photon. 15, 837 (2021).
  4. O. Dogadov, C. Trovatello, B. Yao, G. Soavi, and G. Cerullo, Parametric nonlinear optics with layered materials and related heterostructures, Laser Photon. Rev. 16, 2100726 (2022).
  5. P. Herrmann, S. Klimmer, T. Lettau, T. Weickhardt, A. Papavasileiou, K. Mosina, Z. Sofer, I. Paradisanos, D. Kartashov, J. Wilhelm, and G. Soavi, Nonlinear valley selection rules and all-optical probe of broken time-reversal symmetry in monolayer WSe2, Nat. Photon. 19, 300 (2025).
  6. J. A. Wilson and A. D. Yoffe, The transition metal dichalcogenides: Discussion and interpretation of the observed optical, electrical and structural properties, Adv. Phys. 18, 193 (1969).
  7. M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Loh, and H. Zhang, The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets, Nat. Chem. 5, 263 (2013).
  8. S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, 2D transition metal dichalcogenides, Nat. Rev. Mater. 2, 17033 (2017).
  9. N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, Second harmonic microscopy of monolayer MoS2, Phys. Rev. B 87, 161403(R) (2013).
  10. L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, Observation of intense second harmonic generation from MoS2 atomic crystals, Phys. Rev. B 87, 201401(R) (2013).
  11. J. Zhang, W. Zhao, P. Yu, G. Yang, and Z. Liu, Second harmonic generation in 2D layered materials, 2D Mater. 7, 042002 (2020).
  12. L. Du, Z. Huang, J. Zhang, F. Ye, Q. Dai, H. Deng, G. Zhang, and Z. Sun, Nonlinear physics of moiré superlattices, Nat. Mater. 23, 1179 (2024).
  13. W. Huang, Y. Xiao, F. Xia, X. Chen, and T. Zhai, Second harmonic generation control in 2D layered materials: Status and outlook, Adv. Funct. Mater. 34, 2310726 (2024).
  14. W.-T. Hsu, Z.-A. Zhao, L.-J. Li, C.-H. Chen, M.-H. Chiu, P.-S. Chang, Y.-C. Chou, and W.-H. Chang, Second harmonic generation from artificially stacked transition metal dichalcogenide twisted bilayers, ACS Nano 8, 2951 (2014).
  15. S. Psilodimitrakopoulos, L. Mouchliadis, I. Paradisanos, G. Kourmoulakis, A. Lemonis, G. Kioseoglou, and E. Stratakis, Twist angle mapping in layered WS2 by polarization-resolved second harmonic generation, Sci. Rep. 9, 14285 (2019).
  16. Y. Wang, J. Xiao, S. Yang, Y. Wang, and X. Zhang, Second harmonic generation spectroscopy on two-dimensional materials, Opt. Mater. Express 9, 1136 (2019).
  17. L. Zhou, H. Fu, T. Lv, C. Wang, H. Gao, D. Li, L. Deng, and W. Xiong, Nonlinear optical characterization of 2D materials, Nanomaterials 10, 2263 (2020).
  18. S. Psilodimitrakopoulos, A. Orekhov, L. Mouchliadis, D. Jannis, G. M. Maragkakis, G. Kourmoulakis, N. Gauquelin, G. Kioseoglou, J. Verbeeck, and E. Stratakis, Optical versus electron diffraction imaging of twist-angle in 2D transition metal dichalcogenide bilayers, npj 2D Mater. Appl. 5, 77 (2021).
  19. I. Paradisanos, A. M. S. Raven, T. Amand, C. Robert, P. Renucci, K. Watanabe, T. Taniguchi, I. C. Gerber, X. Marie, and B. Urbaszek, Second harmonic generation control in twisted bilayers of transition metal dichalcogenides, Phys. Rev. B 105, 115420 (2022).
  20. R. W. Boyd, Nonlinear Optics, 4rd ed. (Academic Press, London, 2020).
  21. J. Shi, P. Yu, F. Liu, P. He, R. Wang, L. Qin, J. Zhou, X. Li, J. Zhou, X. Sui, S. Zhang, Y. Zhang, Q. Zhang, T. C. Sum, X. Qiu, Z. Liu, and X. Liu, 3R MoS2 with broken inversion symmetry: A promising ultrathin nonlinear optical device, Adv. Mater. 29, 1701486 (2017).
  22. C. Attaccalite and M. Grüning, Nonlinear optics from an ab initio approach by means of the dynamical Berry phase: Application to second- and third-harmonic generation in semiconductors, Phys. Rev. B 88, 235113 (2013).
  23. C. Attaccalite, M. Palummo, E. Cannuccia, and M. Grüning, Second-harmonic generation in single-layer monochalcogenides: A response from first-principles real-time simulations, Phys. Rev. Mater. 3, 074003 (2019).
  24. N. A. Pike and R. Pachter, Second-order nonlinear optical properties of monolayer transition-metal dichalcogenides by computational analysis, J. Phys. Chem. C 125, 11075 (2021).
  25. J. Ruan, Y.-H. Chan, and S. G. Louie, Exciton enhanced nonlinear optical responses in monolayer h-BN and MoS2: Insight from first-principles exciton-state coupling formalism and calculations, Nano Lett. 24, 15533 (2024).
  26. M. Grüning and C. Attaccalite, Second harmonic generation in h-BN and MoS2 monolayers: Role of electron-hole interaction, Phys. Rev. B 89, 081102(R) (2014).
  27. C. Wang, X. Liu, L. Kang, B.-L. Gu, Y. Xu, and W. Duan, First-principles calculation of nonlinear optical responses by Wannier interpolation, Phys. Rev. B 96, 115147 (2017).
  28. P. Garcia-Goiricelaya, J. Krishna, and J. Ibañez-Azpiroz, Including many-body effects into the Wannier-interpolated quadratic photoresponse tensor, Phys. Rev. B 107, 205101 (2023).
  29. K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin MoS2: A new direct-gap semiconductor, Phys. Rev. Lett. 105, 136805 (2010).
  30. G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).
  31. G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B 49, 14251 (1994).
  32. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  33. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)], Phys. Rev. Lett. 78, 1396 (1997).
  34. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  35. S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132, 154104 (2010).
  36. A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178, 685 (2008).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ldws-7yvc for k-mesh convergence tests of the χijk(2)(ω) spectra (Sec. I) and validation of the Wannier-interpolated band structures against DFT for all studied configurations (Sec. II).
  38. J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000).
  39. A. Kormányos, V. Zólyomi, N. D. Drummond, P. Rakyta, G. Burkard, and V. I. Fal'ko, Monolayer MoS2: Trigonal warping, the Γ valley, and spin-orbit coupling effects, Phys. Rev. B 88, 045416 (2013).
  40. A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Emerging photoluminescence in monolayer MoS2, Nano Lett. 10, 1271 (2010).
  41. D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides, Phys. Rev. Lett. 108, 196802 (2012).
  42. X. Bao, S.-Q. Li, H. Chen, Q. Chen, Y. Wu, L. Zhang, Y. Xie, and Y. Chen, Extremely anisotropic second harmonic generation in in-plane Janus transition metal dichalcogenide heterostructures, Phys. Rev. B 114, 045418 (2026).
  43. S. Patra and C. Cocchi, Wannier-based second harmonic generation (SHG) data for monolayer and different bilayer stackings of MoS2 [Data set], Zenodo, 2026, https://doi.org/10.5281/zenodo.21480888.
  44. C. Xu, Y. Mao, K. Li, Z. Wang, X. Liu, N. Dong, S.-Y. Li, A. Pan, and J. Wang, Nonlinear optical response modulated by lattice reconstruction in small-angle twisted bilayer MoS2, Adv. Opt. Mater. 12, 2302407 (2024).

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Phys. Rev. B 114, 225412– Published 19 October, 2026
  • Received 22 July 2026
  • Revised 25 September 2026
  • Accepted 28 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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